## Stochastic Cooling Techniques: A new tool for heavy-ion collisions?

Beam cooling is a method of reducing the energy spread, the angular divergence and the geometric size of a charged-particle beam that circulates in an accelerator without causing any beam loss. Particles are “compressed” in the beam, and the particle density increases, resulting in a higher number of particles participating in the collisions.

Beam size varies, as particles are oscillating around the beam centre. The mean square velocity spread is used to define the beam temperature in analogy to the temperature of the gas. By analogy with thermodynamics, the term “cooling” denotes the reduction of the disorder in the beam (i.e. by analogy with the kinetic theory of gases, where heat is described as disorder in the motion of the gas molecules). The larger the mean square of the velocity of these oscillations in a beam is, the larger its size. The main aim of cooling is to compress the same number of particles into a beam of smaller size and energy spread, maintaining their energies close to a mean value.

In one of his often cited papers (”The Status of Stochastic Cooling”), Mohl (1997) discussed three reasons that demonstrate the importance of the beam quality, achieved by applying cooling techniques. First, cooling results in a higher phase-space, available for the accumulation of rare particles (for example the Antiproton Accumulator at CERN). Second, it provides sharply collimated highly mono-energetic beams that can be used for precision experiments in colliding beams and in fixed target experiments; thus giving a higher luminosity. Finally, cooling is important for retaining the beam quality, as it compensates for various mechanisms that lead to the growth of the beam size like the intra-beam scattering (IBS) or other phenomena that need to be taken into account in collisions.

Two types of beam cooling have been demonstrated and used at various laboratories: electron cooling, which was pioneered by G. I. Budker and associates at Novosibirsk, and stochastic cooling, developed by Simon van der Meer of CERN (S. van der Meer, 1985; 1994). Electron cooling acquired its name from the fact that an electron beam is used to cool the particles by removing energy. Stochastic cooling is named after the stochastic nature of the beam – i.e., particles moving randomly with respect to one another. The development of stochastic cooling introduced a new experimental regime and led to the discovery of the W and Z bosons.

Stochastic cooling is a feedback system comprising two components: a detector or pick up electrode, which senses the motion of the particles, and a corrector or the kicker, which deflects the particle by an angle proportional to its error and hence adjusts the particle's angles. The detector produces an error signal, with amplitude proportional to the particle's deviation from the central orbit at the pick-up. This signal is subsequently amplified and applied to the kickers. In short, the pick-up detects the deviations of the particles from the centre with respect to the requisite orbit and the kicker provides a corrective angular kick. Their separation is chosen to correspond to a quarter of the betatron oscillation wavelength (plus an integer number of the half wavelength). To obtain the optimal correction, the pick-up should be placed at the peak of the oscillation amplitude, while the kicker should be chosen to sit at the zero passage.

Stochastic cooling: As particles travel around a circular accelerator, a detector or "pick up" measures their motion and sends a signal across the ring to a corrector, the kicker, which adjusts their angles (Image: CERN)

Single-particle model for a transverse stochastic cooling system.

*Image from taken from Fermilab - Accelerator Division Document Databases*

In an ideal world the particle trajectories would move even closer to the central orbit. However, we are not dealing with a single particle but with a bunch of particles that come at slightly different positions. As a result, not all of them cross the pick-up at the crest of its oscillations. This means that some of them receive a partial correction, hence the process has to be repeated several times in order to get a refined result. In fact, cooling systems require many beam revolutions to cool the beam due to the large number of particles involved and the finite bandwidth of the hardware. A sufficient amount of time and a number of circulations are needed to achieve the ideal result.

*Optimum spacing between pick-up and kicker.*

Another important aspect of stochastic cooling is the time that the signal needs to travel from the pick-up to the kicker. For optimum cooling the signal should arrive at exactly the same time as the particles. A delay is induced from the fact that the signal is first amplified, and, moreover, is travelling through a cable with a speed lower than the speed of the particles, which is close to the speed of light. What's the solution to this problem? A shortcut can be created for the correction signal, so that it will have to travel a shorter distance from the pick-up to the kicker. This allows the correcting signal to reach the kicker just on time to be applied to the beam. However, for reasons mentioned above, this process still needs a number of iterations and we don't expect to have the perfect cooling during the first passage of the beam.

As explained, the cooling techniques are based on the “pick up” of different samples of particles from the beam. The term "stochastic cooling" is derived from the need for a random or stochastic sample of particles passing through the pickup upon each revolution for cooling to work effectively. Although they are all based on the same principles, it should be noted that today there are many different stochastic cooling schemes built in different accelerator machines around the world. They depend on the specificities of the machines, the accelerated species, the desired cooling etc. A wide variety of pickup designs, power amplifiers and filter techniques have been used in order to improve the performance of stochastic cooling systems (Goldberg et al, 1990; Goldbert & Lambertson, 1992).

The first practical cooling system was demonstrated at the ISR, while early R&D work was done in storage rings dedicated to that purpose like ICE at CERN and the 200 MeV electron cooling ring (ECR) at FNAL.

These R&D efforts proved that stochastic cooling could apply to all three axes (horizontal, longitudinal and vertical). Indeed, today stochastic cooling systems are based on 3D information, which significantly improves the efficiency of this technique. The critical demonstration of simultaneous stochastic beam cooling in all three planes enabled the confident construction of the CERN Antiproton Accumulator (AA) and later the two-ring Fermilab antiproton source including the Debuncher and the Accumulator Rings (Marriner, 2003).

## Stochastic Cooling for Heavy Ions

Stochastic cooling is an advanced technique that has been successfully applied to create intense particle sources and to reduce beam size, especially in the energy regime where electron cooling has traditionally been judged to be more difficult. Recently, new applications of stochastic cooling for colliding heavy-ion beams at RHIC and for modest energy proton beams have appeared.

The applications of stochastic cooling in these new fields open up new opportunities for the development of the LHC. Currently, IBS influences the dynamics of high intensity lead beams significantly. As a result, emittance growth and particle losses occur. In the heavy ion runs, where 3 experiments will be collecting data, the average fill duration will be rather short due to the high burn of rate. However, these issues could be improved, if stochastic cooling is used, reducing the emittance growth and the debunching component during injections and collisions.

* The author would like to thank John Jowett and Michaela Schaumann for the sources they offered me and the discussions that we had in the process of writing this article. *

* Further readings *

D. Mohl, G. Petrucci, L. Thorndahl, and S. van der Meer, Phys. Rept. 58 (1980), 76.

D. Mohl, Nucl. Instrum. & Meth. A391 (1997), 164.

D. M?hl, Stochastic Cooling for Beginners, PS/LEA/Note 84-12.

John Marriner, David McGinnis, An Introduction to Stochastic Cooling, AIP Conference Proceeding #249 VI, 693-761 (1990)