Radioactive Decay


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^- + ν_e

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⁺ + ν_e

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 + ν_e

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)

Radioactive Half Life:

N₀/2 = N₀exp(-λt_1/2)

∴ 1/2 = exp(-λt_1/2)

∴ 2 = exp(λt_1/2)

∴ ln2 = λt_1/2

∴ t_1/2 = (ln2)/λ

∴ t_1/2 = 0.693/λ

Define mean lifetime as τ = 1/λ

∴ t_1/2 = 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.

Radiative Effects
-----------------

When passing through matter, a β particle can be decelerated by 
electromagnetic interactions with other charged particles, typically 
other electrons or by an atomic nucleus, and emit BREMSSTRAHLUNG 
X-RAYS. The kinetic energy lost by the particle is converted into 
a photon with an equivalent energy.

In water, β particles typically exceed the speed of light
(which is 75% that of light in vacuum), and thus generate blue 
CHERENKOV RADIATION.  This process is analagous to a sonic boom
where the disturbance created by the β particle cannot relax
back to equilibrium forming a shock front that radiates energy.

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