KAHIBARO
Table of Contents
Energy as a Conserved Quantity
Energy is one of the most important conserved quantities in physics. The idea of conservation means that in an isolated system, the total energy does not appear from nowhere and does not disappear. It can change form, and it can move from one part of a system to another, but the total amount remains constant.
In particle and nuclear physics, conservation of energy is used constantly to decide whether a process can happen, to calculate the energies of particles after a reaction or decay, and to understand thresholds for creating new particles. Even when the forms of energy look very different, the same conservation rule applies.
For an isolated system, total energy is conserved.
If a process occurs, then
$$
E_{\text{initial}} = E_{\text{final}}.
$$
What Counts as Energy
In everyday mechanics, energy may appear as kinetic energy or potential energy. In nuclear and particle physics, the bookkeeping is broader. A system can contain rest energy, kinetic energy, binding energy, excitation energy, and radiation energy carried by photons.
A particle of mass $m$ has rest energy
$$
E_0 = mc^2.
$$
If the particle is moving, it also has kinetic energy. In relativistic physics, total energy is not just the classical expression $\frac{1}{2}mv^2$. Instead, energy and momentum are related by
$$
E^2 = p^2 c^2 + m^2 c^4.
$$
This means that mass itself contributes to the energy balance. Because of this, reactions can convert kinetic energy into new particle mass, and mass differences between initial and final states can appear as released kinetic energy or radiation.
Energy in Decays
A decay process is possible only if the total initial energy is at least as large as the total final energy. Consider a particle $A$ decaying into particles $B$ and $C$:
$$
A \to B + C.
$$
If particle $A$ is initially at rest, then its total energy is just its rest energy:
$$
E_{\text{initial}} = m_A c^2.
$$
The final energy is the sum of the total energies of $B$ and $C$:
$$
E_{\text{final}} = E_B + E_C.
$$
For the decay to occur,
$$
m_A c^2 = E_B + E_C.
$$
If $m_A$ is larger than $m_B + m_C$, the extra energy appears as kinetic energy of the decay products. If $m_A$ is smaller than $m_B + m_C$, the decay cannot happen spontaneously.
A spontaneous decay requires that the initial rest energy be large enough to supply the final rest energies and any kinetic energy:
$$
m_{\text{initial}} c^2 \ge \sum E_{\text{final}}.
$$
Energy in Nuclear Reactions
In nuclear reactions, the same principle applies. Suppose a reaction has the form
$$
a + A \to b + B.
$$
The total energy before the reaction includes the rest energies and kinetic energies of the incoming particles. The total energy after the reaction includes the rest energies and kinetic energies of the outgoing particles.
A useful quantity is the $Q$ value:
$$
Q = \left(m_{\text{initial}} - m_{\text{final}}\right)c^2.
$$
If $Q > 0$, the reaction releases energy. If $Q < 0$, energy must be supplied for the reaction to occur.
This is especially important in nuclear physics because a small change in mass corresponds to a large energy change, due to the factor $c^2$.
Rest Energy and Mass Differences
One of the most striking lessons of modern physics is that mass and energy are deeply connected. If the total mass of the final particles is smaller than the total mass of the initial particles, the difference does not vanish. It becomes energy in another form, often kinetic energy or photon energy.
In nuclear processes, this is closely related to binding energy. A bound nucleus often has less mass than the separate nucleons that make it up. The missing mass corresponds to binding energy.
In particle reactions, newly created particles can be produced if enough energy is available. High kinetic energy in collisions can be transformed into the rest energy of massive particles.
Mass differences correspond to energy differences:
$$
\Delta E = \Delta m \, c^2.
$$
Energy and Radiation
Photons have no rest mass, but they do carry energy. The energy of a photon is
$$
E = hf,
$$
where $h$ is Planck's constant and $f$ is the frequency.
Because photons carry energy, they play a major role in conservation laws. In many nuclear and particle processes, a photon is emitted to carry away exactly the energy needed to satisfy conservation.
For example, an excited nucleus can move to a lower energy state by emitting a gamma ray:
$$
N^* \to N + \gamma.
$$
The photon energy equals the decrease in nuclear energy, apart from any small recoil effects of the nucleus.
Threshold Energy
Some reactions do not occur unless the incoming particles have enough energy. This minimum required energy is called threshold energy. It is needed when the final particles have a larger total rest mass than the initial particles.
For example, in a collision,
$$
a + b \to c + d,
$$
if
$$
m_c + m_d > m_a + m_b,
$$
then some kinetic energy must be converted into rest energy.
This is how particle accelerators create new particles. They supply very large kinetic energies so that collisions can produce heavier particles.
Why Energy Alone Is Not Always Enough
Energy conservation is necessary for every physical process, but by itself it is not always enough to determine whether a process can happen. Other conservation laws, such as momentum, electric charge, baryon number, or lepton number, may also have to be satisfied.
A reaction may satisfy energy conservation and still be forbidden because another conserved quantity fails. Conversely, if a reaction violates energy conservation, it cannot occur, no matter what the other quantities do.
So energy conservation is universal and essential, but it works together with other conservation laws.
Simple Energy Bookkeeping
The following table shows common forms of energy that appear in nuclear and particle physics.
| Form of energy | Typical expression | Meaning |
|---|---|---|
| Rest energy | $mc^2$ | Energy due to mass |
| Kinetic energy | depends on speed, relativistic in general | Energy of motion |
| Photon energy | $E = hf$ | Energy carried by radiation |
| Binding energy | from mass difference | Energy associated with bound systems |
| Excitation energy | difference between internal states | Internal energy above ground state |
Example Idea
Suppose a particle at rest decays into two lighter particles:
$$
A \to B + C.
$$
If
$$
m_A c^2 > (m_B + m_C)c^2,
$$
then the difference
$$
\left(m_A - m_B - m_C\right)c^2
$$
appears as kinetic energy of $B$ and $C$.
This is a standard pattern in radioactive decay and elementary particle decay. The initial rest energy is redistributed into final rest energy plus motion.
Visualizing Energy Before and After
Final Perspective
Energy conservation is one of the most powerful tools in all of physics. In nuclear and particle physics, it allows us to test whether reactions are possible, determine the energies of emitted particles, understand mass differences, and explain how collisions create new matter.
The central rule is always
$$
E_{\text{initial}} = E_{\text{final}}.
$$
In relativistic physics, mass is part of energy, and particle processes must conserve total energy including rest energy, kinetic energy, and radiation energy.