In particle physics, the **threshold energy** for production of a particle is the minimum kinetic energy a pair of traveling particles must have when they collide. The threshold energy is always greater than or equal to the rest energy of the desired particle. In most cases, since momentum is also conserved, the threshold energy is significantly greater than the rest energy of the desired particle - and thus there will still be considerable kinetic energy in the final particles.

The **threshold energy** should not be confused with the threshold displacement energy, which is the minimum energy needed to permanently displace an atom in a crystal to produce a crystal defect in radiation material science.

Consider the collision of a mobile proton with a stationary proton so that a meson is produced:

Transforming into the ZMF (Zero Momentum Frame or Center of Mass Frame) and assuming the outgoing particles have no KE (kinetic energy) when viewed in the ZMF, the conservation of energy equation is:

Rearranged to

By assuming that the outgoing particles have no KE in the ZMF, we have effectively considered an inelastic collision in which the product particles move with a combined momentum equal to that of the incoming proton in the Lab Frame.

Our terms in our expression will cancel, leaving us with:

Using relativistic velocity additions:

We know that is equal to the speed of one proton as viewed in the ZMF, so we can re-write with :

So the energy of the proton must be MeV.

Therefore, the minimum kinetic energy for the proton must be MeV.

Consider the case where a particle 1 with lab energy (momentum ) and mass impinges on a target particle 2 at rest in the lab, i.e. with lab energy and mass . The threshold energy to produce three particles of masses , , , i.e.

is then found by assuming that these three particles are at rest in the center of mass frame (symbols with hat indicate quantities in the center of mass frame):

Here is the total energy available in the center of mass frame.

Using , and one derives that

^{ [1] }

In nuclear physics, **beta decay** (β-decay) is a type of radioactive decay in which a beta particle is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For example, beta decay of a neutron transforms it into a proton by the emission of an electron accompanied by an antineutrino; or, conversely a proton is converted into a neutron by the emission of a positron with a neutrino in so-called *positron emission*. Neither the beta particle nor its associated (anti-)neutrino exist within the nucleus prior to beta decay, but are created in the decay process. By this process, unstable atoms obtain a more stable ratio of protons to neutrons. The probability of a nuclide decaying due to beta and other forms of decay is determined by its nuclear binding energy. The binding energies of all existing nuclides form what is called the nuclear band or valley of stability. For either electron or positron emission to be energetically possible, the energy release or *Q* value must be positive.

An ideal **Fermi gas** is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density, temperature, and the set of available energy states. The model is named after the Italian physicist Enrico Fermi.

**Surface free energy** or **interfacial free energy** or **surface energy** quantifies the disruption of intermolecular bonds that occurs when a surface is created. In the physics of solids, surfaces must be intrinsically less energetically favorable than the bulk of a material, otherwise there would be a driving force for surfaces to be created, removing the bulk of the material. The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces. There is "excess energy" as a result of the now-incomplete, unrealized bonding at the two surfaces.

In physics, a **wave vector** is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important. Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

A **radiation zone**, or **radiative region** is a layer of a star's interior where energy is primarily transported toward the exterior by means of radiative diffusion and thermal conduction, rather than by convection. Energy travels through the radiation zone in the form of electromagnetic radiation as photons.

In rotordynamics, the **rigid rotor** is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the *linear rotor* requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.

The **relativistic Breit–Wigner distribution** is a continuous probability distribution with the following probability density function,

In physics, the **Thomas precession**, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion.

In particle physics, **neutral particle oscillation** is the transmutation of a particle with zero electric charge into another neutral particle due to a change of a non-zero internal quantum number, via an interaction that does not conserve that quantum number. Neutral particle oscillations were first investigated in 1954 by Murray Gell-mann and Abraham Pais.

In electrodynamics, the **Larmor formula** is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.

A **theoretical motivation for general relativity**, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation *a priori*. This provides a means to inform and verify the formalism.

**Plasma parameters** define various characteristics of a plasma, an electrically conductive collection of charged particles that responds *collectively* to electromagnetic forces. Plasma typically takes the form of neutral gas-like clouds or charged ion beams, but may also include dust and grains. The behaviour of such particle systems can be studied statistically.

In relativity, **rapidity** is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates.

In stellar physics, the **Jeans instability** causes the collapse of interstellar gas clouds and subsequent star formation, named after James Jeans. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter. For stability, the cloud must be in hydrostatic equilibrium, which in case of a spherical cloud translates to:

In the theory of general relativity, a **stress–energy–momentum pseudotensor**, such as the **Landau–Lifshitz pseudotensor**, is an extension of the non-gravitational stress–energy tensor that incorporates the energy–momentum of gravity. It allows the energy–momentum of a system of gravitating matter to be defined. In particular it allows the total of matter plus the gravitating energy–momentum to form a conserved current within the framework of general relativity, so that the *total* energy–momentum crossing the hypersurface of *any* compact space–time hypervolume vanishes.

**Particle decay** is the spontaneous process of one unstable subatomic particle transforming into multiple other particles. The particles created in this process must each be less massive than the original, although the total invariant mass of the system must be conserved. A particle is unstable if there is at least one allowed final state that it can decay into. Unstable particles will often have multiple ways of decaying, each with its own associated probability. Decays are mediated by one or several fundamental forces. The particles in the final state may themselves be unstable and subject to further decay.

In probability theory and statistics, the **normal-inverse-gamma distribution** is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.

In physics, **relativistic angular momentum** refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the three-dimensional quantity in classical mechanics.

In physics, the **Maxwell–Jüttner distribution** is the distribution of speeds of particles in a hypothetical gas of relativistic particles. Similar to Maxwell's distribution, the Maxwell–Jüttner distribution considers a classical ideal gas where the particles are dilute and do not significantly interact with each other. The distinction from Maxwell's case is that effects of special relativity are taken into account. In the limit of low temperatures much less than , this distribution becomes identical to the Maxwell–Boltzmann distribution.

- ↑ Jackson, John.
*Classical Electrodynamics*. Wiley. pp. 533–539. ISBN 978-0-471-30932-1.

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