Wolfram Alpha:
Search by keyword:
Astronomy
Chemistry
Classical Mechanics
Classical Physics
Climate Change
Cosmology
Finance and Accounting
Game Theory
General Relativity
Group Theory
Lagrangian and Hamiltonian Mechanics
Macroeconomics
Mathematics
Mathjax
Microeconomics
Nuclear Physics
Particle Physics
Probability and Statistics
Programming and Computer Science
Quantitative Methods for Business
Quantum Computing
Quantum Field Theory
Quantum Mechanics
Semiconductor Reliability
Solid State Electronics
Special Relativity
Statistical Mechanics
String Theory
Superconductivity
Supersymmetry (SUSY) and Grand Unified Theory (GUT)
The Standard Model
Topology
Units, Constants and Useful Formulas
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)
Radioactive Half Life:
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.
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.