The creation and study of new elementary particles is an important part of
contemporary physics. Especially interesting is the discovery of a very
massive particle. To create a particle of mass \(M\) requires an energy \(M
c^{2}\). With enough energy, an exotic particle can be created by allowing a
fast-moving particle of ordinary matter, such as a proton, to collide with a
similar target particle. Let us consider a perfectly inelastic collision
between two protons: An incident proton with mass \(m\), kinetic energy \(K\), and
momentum magnitude \(p\) joins with an originally stationary target proton to
form a single product particle of mass \(M .\) You might think that the creation
of a new product particle, 9 times more massive than in a previous experiment,
would require just 9 times more energy for the incident proton. Unfortunately,
not all of the kinetic energy of the incoming proton is available to create
the product particle, since conservation of momentum requires that after the
collision the system as a whole still must have some kinetic energy. Only a
fraction of the energy of the incident particle is thus available to create a
new particle. You will determine how the energy available for particle
creation depends on the energy of the moving proton. Show that the energy
available to create a product particle is given by
$$
M c^{2}=2 m c^{2} \sqrt{1+\frac{K}{2 m c^{2}}}
$$
From this result, when the kinetic energy \(K\) of the incident proton is large
compared to its rest energy \(m c^{2}\), we see that \(M\) approaches \((2 m K)^{1
/ 2} / c .\) Thus if the energy of the incoming proton is increased by a factor
of 9, the mass you can create increases only by a factor of \(3 .\) This
disappointing result is the main reason that most modern accelerators, such as
those at CERN (in Europe), at Fermilab (near Chicago), at SLAC (at Stanford),
and at DESY (in Germany), use colliding beams. Here the total momentum of a
pair of interacting particles can be zero. The center of mass can be at rest
after the collision, so in principle all of the initial kinetic energy can be
used for particle creation, according to
$$
M c^{2}=2 m c^{2}+K=2 m c^{2}\left(1+\frac{K}{2 m c^{2}}\right)
$$
where \(K\) is the total kinetic energy of two identical colliding particles.
Here, if \(K>>m c^{2}\), we have \(M\) directly proportional to \(K\), as we would
desire. These machines are difficult to build and to operate, but they open
new vistas in physics.
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