## F4    Relativistic Collisions

1. Inelastic Frontal Collision of Two Like Particles, One at Rest

A particle of rest mass m0 collides with v = 12 / 13 • c with a particle at rest of the same type and thereby merges into a new particle of rest mass M0, which after its creation has a velocity u. The conservation of momentum and the conservation of energy-mass must be met:

mv• v + m0• 0 = Mu• u          and         mv• c2 + m0• c2 = Mu• c2

thus        mv• v = Mu• u       and      m0 + mv = Mu together with         mv = m0 / √ = m0 / (5/13) = 2.6 • m0

hence     u = v • (mv / Mu ) = v • (mv / (mo + mv)) = v • ( 1 / (√ + 1) ) = 12/13 • c • (1/(5/13 + 1)) = 2/3 • c

and M0 = Mu • √(1 - (2/3)2) ≈ ( m0 + mv ) • 0.745 ≈ m0 • 3.6 • 0.745 ≈ 2.68 • m0

2. Inelastic Frontal Collision of Two Like Particles, Moving in Opposite Directions

Two particles of rest mass m0 collide head on with v = ± c • 12 / 13 and merge to form a new particle of rest mass M0. We again write the two conservation laws:

mv• v + mv• (-v) = Mu• u       and        mv• c2 + mv• c2 = Mu• c2

thus        0 = Mu• u       and       2 • mv = Mu together with mv = m0 / √ = m0 / (5/13) = 2.6 • m0

hence     u = 0       and      Mu = M0 = 2 • mv = 2 • 2.6 • m0 = 5.2 • m0

The numerical difference of the two is not very impressive. This is only because we have in our example not ‘approached c’. For v -> c, however, the expression v • (1 / (√ + 1)) for u approaches more and more v, which means that the particle produced will also have a speed very close to c, and therefore Mv will be much larger than M0! Today's particle accelerators deliver speeds that are only a few m / s or even cm / s smaller than c! Thus the first method above requires a lot more energy to produce a heavier (possibly hypothetical) particle of a given rest mass, because a larger portion of the input energy is spent on the unavoidable kinetic energy of the particle produced. Only the second method can use the complete input energy to produce the new particle (see problems 4 and 5 in F7).

This is the reason that modern facilities like to be equipped with double storage rings, in which the particles (or particles and anti-particles) race around in opposite directions at speeds close to c, before being brought to frontal collisions inside huge detectors. Such a facility in the vicinity of Hamburg (DESY ~ German Electron Synchrotron) for electrons and positrons has already been active for many years. See the relevant section in the book [11] by Sexl! CERN near Geneva is at the moment (2006) expanding its large plant expressly to handle much heavier protons (LHC ~Large Hadron Collider).

Both the DESY (www.desy.de) and CERN (www.cern.ch) provide informative websites. It was, incidentally, at CERN that Tim Berners Lee developed the Internet in its present form in order to facilitate teams whose members live and work in various corners of the world.

View of the 28 km circular tunnel which lies 100 m below the earth’s surface. In the spring of 2005, superconducting magnets
were installed to hold the protons in their circular path through the two storage rings

ATLAS, one of the four enormous detectors that record the results of the direct collision of the protons. It gathers in a very short
time an amount of data equivalent to the entire European telecommunications network.