Threshold Energy for Particle Production Calculator

Threshold energy is the minimum kinetic (or total) energy a beam particle must have, striking a stationary target, to produce a specific set of final-state particles. Below this energy, the reaction is kinematically forbidden no matter how the collision occurs, energy and momentum conservation simply cannot be satisfied.

E₁ = (M²−m₁²−m₂²)/(2m₂), where E₁ is the beam particle's total energy at threshold, M is the total rest mass of all final-state particles, m₁ is the beam particle's rest mass, and m₂ is the stationary target particle's rest mass.

Threshold is defined as the minimum possible beam energy for the reaction to occur, and the lowest-energy configuration that conserves momentum is for all produced particles to move together as a single lump at the center-of-mass velocity, with zero relative motion between them, this uses the least possible kinetic energy for the given final-state rest mass.

Threshold occurs exactly when the fixed-target center-of-mass energy √s equals the total rest mass M of the final-state particles. This calculator solves the same s=m₁²+m₂²+2E₁m₂ relation used by the Center of Mass Energy Calculator, just rearranged to solve for E₁ given a target √s=M.

Yes, setting m₁=0 correctly handles massless beam particles like photons. This is exactly the calculation used for photoproduction thresholds, such as finding the minimum photon energy needed to produce a pion when striking a stationary proton.

In a fixed-target setup, momentum conservation forces the produced particles to move forward as a group, carrying away kinetic energy that is unavailable for particle production. A symmetric collider has zero net momentum, so far less total beam energy is needed to reach the same final-state mass, this is exactly why modern high-energy physics favors colliders for reaching the highest particle masses.

The Bevatron accelerator at Berkeley was specifically designed with enough beam energy to exceed the threshold for antiproton production (p+p→p+p+p+p̄), about 5.6 GeV of kinetic energy, enabling the 1955 discovery of the antiproton, a landmark confirmation of Dirac's prediction of antimatter.

It reports both: E₁ is the beam particle's total relativistic energy (rest mass energy plus kinetic energy) at threshold, and the kinetic energy is separately shown as E₁ minus the beam particle's rest mass m₁, since kinetic energy is usually the more directly meaningful quantity for accelerator design.

Exactly at threshold, the reaction can occur but produces the final-state particles all moving together with zero relative velocity, essentially 'just barely' happening. Above threshold, there is additional energy available, some of which can appear as relative motion (and hence additional invariant mass headroom) among the final-state particles.

Yes, M represents the total rest mass of however many particles are produced, whether it's two particles (like p+p→p+p+π0) or several (like p+p→p+p+p+p̄, four final-state particles), the formula only needs the sum of their rest masses.