1.1.6: Conservation Laws

The conservation laws state that mass, energy, momentum, charge and matter/antimatter can not be created or destroyed.

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Equivalence of Mass and Energy

An important concept is that mass and energy are equal, through the famous equation:

\begin{equation} E=mc^2 \end{equation}

ie: Energy is equal to the mass of an object (in kg) multiplied by the square of the speed of light.

Conservation of Mass and Energy

Since mass and energy are equivalent, reactions can occur where energy is converted into mass or vice versa.
For instance, in pair production (a photon interaction described in another section) a high energy photon may interact within the electromagnetic field of a nucleus. The photon disappears, but creates an electron and a positron with kinetic energy equivalent to the initial energy of the photon minus the energy required to create the mass of the two leptons.
The opposite is also true. When fission of uranium occurs, the resulting particles do not contain the same mass as the original uranium atom. This is because some of the mass has been converted to energy.

Conservation of Momentum

Momentum must also be conserved during any interaction of particles. This is relatively easy to imagine with actual particles, but becomes more complicated when photons are involved. Photons are also considered to have a momentum, which must be accounted for when considering particle interactions.
In the case of pair production, the momentum of the two particles created by the photon is not enough to account for the initial momentum of the photon. This excess momentum is absorbed by the nearby nucleus. This explains why pair production only occurs in the vicinity of a nucleus.
Triplet production is very similar to pair production, except that it occurs in the vicinity of an electron instead of a nucleus. The extra momentum is absorbed by the electron, which may also gain enough energy to escape its binding energy to the nucleus. In this case, three particles are released - the electron/positron pair generated by the photons energy, and the orbital electron which gains sufficient energy and momentum to escape the binding energy.

Conservation of Charge

When a nucleus has an unstable number of protons and neutrons, it may convert a proton to a neutron or vice versa. This is only possible when the charge is released as an additional particle. In the case of a proton to neutron conversion (loss of +1) the charge is released as a positron. When a neutron is converted to a proton, an electron (-1) must be released to account for the creation of a +1 charge.

Conservation of Matter/Antimatter

In the situation above, an electron is created to accept the charge from a neutron to proton conversion. This reaction is not allowed unless an equivalent mass of antimatter is also created. An antineutrino is also generated to account for the creation of a normal lepton. When a positron is created by proton to neutron conversion, a normal neutrino is created to balance this conservation law.

Practical Examples in Radiotherapy

The events surrounding Pair Production demonstrate these Laws.

  • The Conservation of Mass is obeyed when the photon of 1.02MeV is converted to 2 electronic particles, AND when the positron and electron annihilate to produce two photons of energy 512keV.
  • The Conservation of Charge is obeyed when the 2 electronic particles produced have opposite charges, and then the two annihilating particles have opposite charges.
  • The Conservation of Momentum is obeyed when the particles and photons produced move off in opposite directions (remember that velocity is a vector not a scalar, so equal masses moving in opposite directions have equal but opposite momentum (and so cancel to 0 when added).