# Threshold energy

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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.

## Contents

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.

## Example

Consider the collision of a mobile proton with a stationary proton so that a ${\pi }^{0}$ meson is produced: $p^{+}+p^{+}\to p^{+}+p^{+}+\pi ^{0}$ 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:

$E=2\gamma m_{p}c^{2}=2m_{p}c^{2}+m_{\pi }c^{2}$ Rearranged to

$\gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {2m_{p}c^{2}+m_{\pi }c^{2}}{2m_{p}c^{2}}}$ 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 $c^{2}$ terms in our expression will cancel, leaving us with:

$\beta ^{2}=1-\left({\frac {2m_{p}}{2m_{p}+m_{\pi }}}\right)^{2}\approx 0.130$ $\beta \approx 0.360$ $v_{\text{lab}}={\frac {u_{\text{cm}}+V_{\text{cm}}}{1+u_{\text{cm}}V_{\text{cm}}/c^{2}}}$ We know that $V_{cm}$ is equal to the speed of one proton as viewed in the ZMF, so we can re-write with $u_{cm}=V_{cm}$ :

$v_{\text{lab}}={\frac {2u_{\text{cm}}}{1+u_{\text{cm}}^{2}/c^{2}}}\approx 0.64c$ So the energy of the proton must be $E=\gamma m_{p}c^{2}={\frac {m_{p}c^{2}}{\sqrt {1-(v_{\text{lab}}/c)^{2}}}}=1221\,$ MeV.

Therefore, the minimum kinetic energy for the proton must be $T=E-{m_{p}c^{2}}\approx 280$ MeV.

## A more general example

Consider the case where a particle 1 with lab energy $E_{1}$ (momentum $p_{1}$ ) and mass $m_{1}$ impinges on a target particle 2 at rest in the lab, i.e. with lab energy and mass $E_{2}=m_{2}$ . The threshold energy $E_{1,{\text{thr}}}$ to produce three particles of masses $m_{a}$ , $m_{b}$ , $m_{c}$ , i.e.

$1+2\to a+b+c,$ 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):

$E_{\text{cm}}=m_{a}c^{2}+m_{b}c^{2}+m_{c}c^{2}={\hat {E}}_{1}+{\hat {E}}_{2}=\gamma (E_{1}-\beta p_{1}c)+\gamma m_{2}c^{2}$ Here $E_{\text{cm}}$ is the total energy available in the center of mass frame.

Using $\gamma ={\frac {E_{1}+m_{2}c^{2}}{E_{\text{cm}}}}$ , $\beta ={\frac {p_{1}c}{E_{1}+m_{2}c^{2}}}$ and $p_{1}^{2}c^{2}=E_{1}^{2}-m_{1}^{2}c^{4}$ one derives that

$E_{1,{\text{thr}}}={\frac {(m_{a}c^{2}+m_{b}c^{2}+m_{c}c^{2})^{2}-(m_{1}c^{2}+m_{2}c^{2})^{2}}{2m_{2}c^{2}}}$ ## Related Research Articles In physics, the kinetic energy (KE) of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest. In physics, the Maxwell–Boltzmann distribution is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. A cyclotron accelerates charged particles outwards from the center along a spiral path. The particles are held to a spiral trajectory by a static magnetic field and accelerated by a rapidly varying electric field. Lawrence was awarded the 1939 Nobel prize in physics for this invention. Bremsstrahlung, from bremsen "to brake" and Strahlung "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into radiation, thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum, which becomes more intense and whose peak intensity shifts toward higher frequencies as the change of the energy of the decelerated particles increases. Synchrotron radiation is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.

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