Beam Cooling
To stand a chance of producing a rare particle among the many events which occur when
the beams collide, many particles must be crowded into a very narrow beam. (See sidebar
on Luminosity in Chapter VI.) In Chapter VI.2 we describe how difficult it is for the
synchrotron rings, which prepare the beam for a collider, to make a beam that is dense
enough. One may lay down beams side by side in a synchrotron, but each will occupy a
different band of transverse or longitudinal energies, rather like streams of amateur
drivers swerving from side to side in their lanes on a motorway. A theorem developed in
the early 1800s by Liouville states that as long as the individual cars can be thought of as
a continuous stream of traffic and as long as the motion is energy conserving (technically
“Hamiltonian”), the traffic cannot be merged; i.e., the beams of particles cannot be
superimposed. All one can do is to “raise driving standards” to reduce the width of each
band of energy and squeeze the streams closer. However, it is possible to circumvent
Liouville’s theorem by considering the individual particles or by having non-energy
conserving processes. The resulting increase in particle (technically “phase space”)
density is called “cooling.” Thus cooling is a process which reduces the energy of a
particle oscillation in either one of three possible directions — horizontal, vertical, or in
the direction of acceleration — to bring the width of the spread in energy among the
particles within the band and bring the bands closer. It is called “cooling” because we can
express the kinetic energy of these oscillations, like the energy of molecules in a gas, by
an equivalent temperature — the hotter the gas, the more energetic is the motion.
A simple, but very effective, cooling mechanism is the radiation of photons from
a circulating beam of electrons. (The beam must be accelerated — in this case by moving
about a circle — in order for it to radiate.) The motion of the electrons alone is NOT
energy-conserving. The complete system of electrons and photons IS energy conserving,
and Liouville’s theorem applies, but if one looks only at the electrons — we don’t care
about the photons — then energy is not conserved and, in fact, the electrons can be made
to cool. Radiation cooling in special “damping rings” is at the heart of linear colliders.
In another sidebar in Chapter VI we describe stochastic cooling — perhaps the
most famous method of cooling because it was used for the Nobel prize-winning
discovery of the W and Z. It works by using the granular nature of a beam of particles,
i.e., that the particles are not really a continuous stream (but almost so). Unlike radiation
cooling, this method works also for heavy particles, like protons.
There are a number of other ingenious methods of cooling. As with the above
methods, each is either special, or particularly effective, for particular particles at
particular energies. One of these — electron cooling — was first developed in 1967 at
Novosibirsk under the aegis of Budker (see sidebar for Budker in Chapter VI). A beam of
heavy particles, antiprotons or ions for example, travels along the same path as a beam of
electrons with the same velocity, sharing its surplus energy with the electrons. There is a
principle of physics called “equipartition of energy” that equalizes the kinetic energy of
the two species of particles through the forces of their mutual electromagnetic fields.
After this equipartition is over, the anti-protons, being almost 2000 times heavier than the
electrons, will only retain about one fiftieth of the velocity spread of their lighter
companions and, if the heavier particle originally had the lion’s share of the energy, it
will be “cooled” by this factor of 50. At high intensity of proton or ion beams this kind of
cooling is a much faster process than stochastic cooling, and is particularly effective
when the proton, ion or antiproton beam is circulating at low energy. At first it was
considered as a rival candidate to the stochastic method of cooling antiprotons for use in
colliders, but stochastic cooling turned out to be much more useful at higher energies,
large emittances and for a relatively small number of antiprotons. Recently electron
cooling has been employed at the high-energy anti-proton recycler at Fermilab and it is
being developed for high-energy ions at RHIC (Brookhaven).
Laser cooling is another method and is also a fascinating practical application of
modern physics. The idea is applied to a beam of ions circulating in a storage ring. The
laser excites the ion from one direction, and the ion then radiates in an isotropic manner
and so cools. (The violation of assumptions for the Liouville theorem is very similar to
that in radiation cooling.) However the radiation must be exactly to the initial state, as the
process must be repeated a million times to cool any one ion.
Ions speeding towards the laser beam see the laser beam shifted in frequency
and hence in photon energy — the Doppler effect. An ion of a particular and precise
velocity can see photons whose energy exactly matches the difference in its electron
energy levels. When such a resonance occurs, energy is transferred between the photon
and the ion, speeding it up or slowing it down. The trick is to sweep the laser frequency,
or the photon energy, backwards and then forwards into the distribution of velocities in
the beam, squeezing it from both sides into a much narrower band. An alternative is to
sweep the particle beam energy — often done with an induction unit.
The separation of the quantum energy levels of the electrons orbiting the ions
must be the same as the energy of the laser photons; and, most importantly, have only a
ground state and one excited state connected by the laser photon. Only a few ions have
these characteristics, but they have been cooled most effectively by this method.
Yet another method is used for the beams for muon colliders. Muons offer a
route to extend circular lepton accelerating storage rings to energies of several TeV, but
the acceleration and storage must occur in a few milliseconds to compete with their rapid
decay. Like antiprotons, the muons must be cooled. Because of the time constraints,
single-pass cooling or perhaps cooling over a few passes, is the only solution. In the socalled “ionization cooling” method, particles pass through an energy absorbing plate and
lose momentum in the direction of their trajectory to the electrons of the absorber
material. (Thus the assumption of energy conservation, in Liouville’s theorem is
violated.) If they are “hot” and therefore moving at a small angle to the axis of the beam,
some of the transverse as well as longitudinal momenta will be lost. An RF cavity
following the absorber replaces the longitudinal but not the transverse momentum.
Repeated a few times, the process will cause a steady reduction of transverse momentum
— reducing the angle of the path and cooling the beam. Longitudinal cooling can also be
accomplished; it requires bending magnets. This approach is being actively studied at this
time. Yet another method is used for the beams for muon colliders. Muons offer a route
to extend circular lepton accelerating storage rings to energies of several TeV, but first
the muons like antiprotons must be cooled, but in a time of a few milliseconds in order to
competes with their rapid decay, with half life 2 microseconds. Single pass cooling, or
perhaps cooling over a few passes, is the only solution. In the so-called “ionisation
cooling” method particles pass through an energy absorbing plate and lose momentum in
the direction of their trajectory to the electrons of the absorber material. (Thus the
assumption of energy conserving, in Liouville’s theorem, is violated.) If they are “hot”
and therefore moving at a small angle to the axis of the beam both some of the transverse
as well as longitudinal momenta will be lost. An RF cavity following the absorber
replaces the longitudinal momentum but not the transverse. Repeated a few times, the
process will cause a steady reduction of transverse momentum – reducing the angle of the
path and cooling the beam. Longitudinal cooling can also be accomplished; it requires
bending magnets. This approach is being actively studied at this time.