Knocking on Heaven's Door - Lisa Randall [54]
Fixed-Target Setup
Particle Collider
[ FIGURE 21 ] Some particle accelerators generate interactions between a beam of particles and a fixed target. Others collide together two particle beams.
However, beam-beam collisions have one big advantage. These collisions can achieve far higher energy. Einstein could have told you the reason that colliders are now favored over fixed-target experiments. It has to do with what is known as the invariant mass of the system. Although Einstein is famous for his theory of “relativity,” he thought a better name would have been “Invariantentheorie.” The real point of his quest was to find a way to avoid being misled by a particular frame of reference—to find the invariant quantities that characterize a system.
This idea is probably more familiar to you for spatial quantities such as length. Length of a stationary object doesn’t depend on how it is oriented in space. An object has a fixed size that has nothing to do with you or your observations, unlike its coordinates, which depend on an arbitrary set of axes and directions you impose.
Similarly, Einstein showed how to characterize events in a way that doesn’t depend on an observer’s orientation or motion. Invariant mass is a measure of total energy. It tells you how massive an object can be created with the energy in your system.
To determine the amount of invariant mass, one could ask this instead: if your system were sitting still—that is, if it had no overall velocity or momentum—how much energy would it contain? If a system has no momentum, Einstein’s equation E = mc2 applies. Therefore, knowing the energy for a system at rest is equivalent to knowing its invariant mass. When the system is not at rest, we need to use a more complicated version of his formula that depends on the value of momentum as well as energy.
Suppose we collide together two beams with the same energy and equal and opposite momentum. When they collide, the momenta add up to zero. That means that the total system is already at rest. Therefore, all the energy—the sum of the energy of the particles in the two individual beams—can be converted to mass.
A fixed-target experiment is very different. One beam has large momentum, but the target itself has none. Not all the energy is available to make new particles because the combined system of the target and the beam particle that hit it is still moving. Because of this motion, not all the energy from the collision can be transferred into making new particles, since some of the energy remains as kinetic energy associated with the motion. It turns out that the available energy scales only with the square root of the product of the energy of the beam and the target. That means, for example, that if we were to increase the energy of a proton beam by 100 and collide it with a proton at rest, the energy available to make new particles would increase by only a factor of 10.
This tells us there is a big difference between fixed-target and beam-beam collisions. The energy of a beam-beam collision is far greater—much bigger than twice as big as a beam-target collision, which is perhaps what you might assume. But that guess would be based on Newtonian thinking, which doesn’t apply for the relativistic particles in that beam that travel at nearly the speed of light. The difference in net energy of fixed-target compared with beam-beam collisions is much bigger than the simple guess because at near the speed of light, relativity comes into play. When we want to achieve high energies, we have no choice but to turn to particle colliders, which accelerate two beams of particles to high energy before colliding them together.