Re: Reason of limit of beam current in high energy accelerators ?

Question:

Reason of limit of beam current in high energy accelerators ?

From: S. Yuri
Grade: Undergraduate
City: Ashqelon State/Prov.: None Country: Israel
Area: Physics
Message ID Number: 958207715.Ph

Why are beam currents in, more in particular, electron - positron colliders and similar installations only reach values of a few hundred mA. Is it because higher beam current values are simply not needed for research, is it due to limited values of power available, limited values of outer pressure on manipulating magnets, safety reasons in case of some containment failure or for some other reason ? Thank you !

Answer:

Because the number of particle interactions, or "events", seen by a particle accelerator is proportional to the beam current, usually the average current is made as high as possible. The beam current in fixed-target accelerators, or the "luminosity" in colliders (which is proportional to the product of the currents in each of the two colliding beams times a focusing factor, due to the precision with which collisions are achieved), are ordinarily second only to beam energy as the most important parameters specifying the capability of the machine. I should mention one caveat though: limitations on the "event rate", the number of events per second. Especially when the beam is accelerated in bunches, as it usually is, events in high-luminosity machines can come so quickly that they cannot be analyzed one-at-a-time, which is the basic method of analysis universal in particle physics. The detector electronics have a finite speed, and if too many events come too quickly, things get seriously confused. Sometimes event rate is the limiting factor in practice, and the beam intensity may have to be reduced to take account of it.

Although I have had only a little experience in particle physics, now long outdated and largely forgotten, I will tell you what I think I know. Aside from the event rate consideration, the accelerator current limit is most fundamentally due to a physical problem, familiar since vacuum tube days, known as "space charge". At the simplest level, it is easy to understand: the particles in the beam, electrons, say, all have the same charge, q, so they repel one another, and thus try to move away from the beam. If this tendency is not controlled, they will hit the walls of the beam pipe and be lost.

The current J in the beam is the charge passing a given point per second, and is thus proportional to the number n of particles per cm³:

where A is the area of the beam and v is its velocity. The factor in parentheses is the number of particles passing by per second. If the density of particles is high enough, the so-called vacuum in the accelerator is actually occupied by a significant charge density -- hence the old term "space charge".

This space charge creates an electric field acting to push the particles away from the regions where the density is highest. We can treat the beam in a particle accelerator as analogous to a beam of light in an optical system. Then Humphries (1999) below shows that the charge of the beam itself can act like a negative lens, tending to defocus the beam. Fortunately, certain types of magnets, known as quadrupoles, can be configured in pairs to act as positive lenses. By use of such positive magnetic lenses, it is possible to prevent the space charge from destroying the beam altogether, by adding such strong positive lenses that the net effect, summed around the accelerator ring, is positive. Then one obtains equations for the transverse motion which are essentially harmonic oscillator equations. Especially in the high energy sections of the accelerator (see below), other disruptive effects come into these equations besides space charge, which are actually more important at limiting the current. Some of these are discussed in the references, below. In any event, with a limited number and strength of quadrupole focusing magnets, one must accept a finite transverse size of the beam.

This is bad enough for a fixed-target accelerator, because the diameter of the beam, and of the beam pipe, has to be increased as a result. Since the price of the magnets, vacuum system, etc, is pretty much proportional to the area of the beam, achieving high current is immediately connected to the cost of the accelerator. It is the designers' job to maximize the beam current constrained by the limitation on maximum cost. However, for colliding beams the situation is more complicated, because in order to get a significant number of collisions between two intersecting particle beams, it will not do to have them spread out laterally, transverse to the direction of motion. You have to concentrate them, that is focus the beam, down as nearly to a point as possible. So at the focus, A is necessarily going to be tiny, and n has to be large.

The v factor in the current tells us that, for a given J, the problems will be most severe early in the acceleration process, since

All the quantities on the right are constant during the acceleration, so this means when v is small, at injection, (n A) must be large in proportion. You can see this exact same effect on a freeway -- as traffic goes faster, cars can be separated more widely, for a given "current" in cars per minute. As a result, high-energy circular accelerators are typically built in stages, with larger aperture (large A) injectors, and the smallest possible beam pipe in the highest energy "main ring", where the overriding cost of the bending magnets is driven by the two factors of aperture and the momentum of the beam particles.

For high-energy modern accelerators, I believe an interesting relativistic effect must help out a great deal in the high-speed portion of the machine. The particle density n we measure in the lab, and which gives us J, is not the same as that seen by an observer riding along with the particles themselves, because of the relativistic contraction of lengths along the beam direction. But it is the density in the particles rest frame that they see as the source of the space charge. The relativistic length contraction factor G is

where E is the particle energy in the beam and E₀ = mc˛ is the particle rest energy. For a 1 TeV proton collider this gives us a factor of 1000, and in each of the two beams. For electrons it is even higher; eg, at 100 GeV, over 3,000,000 in each beam. These large effects probably explain why space charge is generally no longer the dominant source of transverse instability in the high energy sections of current machines. For electron colliders, Ref 2 indicates quantum fluctuations are the dominant cause of transverse disturbance.

As you can see, it is a complex subject overall, with a number of effects coming into play at various stages of the acceleration cycle.

REFERENCES:

This is just a collection of some places you may find interesting for further information.

1. Humphries, S., 1999, Principles of Charged Particle Acceleration , Wiley. A complete textbook, available on-line.

2. Particle Data Group, "Accelerator physics of colliders", in the Review of Particle Properties, the standard particle physics compilation of reference information. Points out that for electron colliders, space charge (read "source properties", for protons) is unimportant in the transverse dynamics at high energy. The chapter on "High-energy collider parameters" gives characteristics of current machines.

3. Beam Physics and Accelerator Technology is a collection of links at SLAC.

4. "Accelerator Physics" is a page maintained by John Jowett of CERN, an especially good source on LEP.

5. Accelerator School 1999. No information in the syllabus, but has a nice list of textbooks on the subject.

6. Scott Snyder (1995), "Thesis" a PhD thesis about the top quark. As is often the case with theses, provides a little tutorial on the Fermi Lab Tevatron collider.

7. This is just for fun.