The trajectories of moving charged particles can be altered through the use of electromagnetic fields. In this manner, particle beams are guided around a circular accelerator, such as a cyclotron or synchrotron, for repetitive encounters with an accelerating cavity. Likewise, beams of particles can be transported from one accelerator to another or from an accelerator toward an experimental target, or even to a patient in medical applications. In each case, this is accomplished by an arrangement of appropriate electromagnetic fields.
Armed with an understanding of the motion of charged particles in magnetic fields, one can imagine a system of electromagnets used to guide a beam of charged particles. Steering a particle from one point to another is only one issue; keeping a stream of charged particles focused along the central trajectory is a vital concern for any beam transport system. Dipole and quadrupole magnets are used for these two purposes. The ability to perform fine adjustments to the particle beam's trajectory and focusing characteristics must also be included in any beam transport system design. For example, smaller, adjustable electromagnets can be used to adjust the position of a particle beam to a small fraction of a millimeter in a particle accelerator that may be many kilometers in circumference. Transport systems can be built to guide and focus particle beams of very high energies to great precision.
Charged Particle Motion in Electromagnetic Fields
The Lorentz force governs the motion of a charged particle,
where E and B are the electric and magnetic field strengths, and q and v are the particle's charge and speed, respectively. The arrows indicate that the direction of the fields and of the particle's motion dictates the direction of the resulting force.
Imagine a positively charged particle entering a localized region of electric field, where the field lines are perpendicular to the initial direction of motion. The particle's trajectory will be deflected in the direction of the electric field lines. The trajectory through this region is a parabola.
Now, suppose a particle enters a localized region of magnetic field where, again, the field lines are perpendicular to the initial direction of motion. The particle's trajectory will be deflected in a direction perpendicular to the particle's velocity and perpendicular to the direction of the magnetic field. Through this region the trajectory will be an arc of a circle.
These two cases are illustrated in Figure 1. By using localized regions of electric and magnetic fields, charged particles can be steered in any general direction, transporting them toward an experimental apparatus, or from one particle accelerator to another, or toward the phosphorescent screen of a television set!
From the Lorentz equation above, it can be seen that particles with very low speeds are more easily governed by electrical forces, whereas particles at very high speeds—say, approaching the speed of light—are more easily affected by magnetic forces.
For comparison, consider a charged particle moving near the speed of light (3 × 108 m/s). If it encounters a magnetic field of strength 1 Tesla (T), which is typical of an iron electromagnet, then the product of these two quantities is 3 × 108 T-m/s, or equivalently, 3 × 108 volt/m. Thus, to generate the same force with an electric field, the electric field strength would have to generate 3 million volts over a distance of 1 cm—an extremely large voltage over a relatively short distance! For particles with high momentum, therefore, magnetic fields are used in beam transport systems.
The ratio of a particle's charge to its momentum is called the magnetic rigidity of the particle and has units of Tesla-meters (T-m). Since the common unit of energy of an elementary particle is the electron volt (eV), and the unit of momentum is written in terms of electron volts divided by the speed of light (eV/c ), then a particle's magnetic rigidity can be conveniently approximated as
where pGeV/c is the particle's momentum in units of GeV/c . (Note: 1 GeV = 109 eV.) The approximation lies in the 3 of the denominator, which comes from the approximation that the speed of light is 3 × 108 m/s. As an example, a particle with a momentum of 3 GeV/c will have a magnetic rigidity of 10 T-m. This says that if the particle is in a magnetic field of 2 T, then it will move in a circular path of radius 5 m. Going one step further, if the particle is in this 2-T field over a distance of only 10 cm, then its trajectory will be deflected through an angle of 10 cm/ 5 m = 0.02 radians (1.15 degrees). The above discussion illustrates how one can build a system of magnetic elements of given lengths and field strengths to steer the paths of particles of a given energy.
The Need for Transverse Focusing
In the design of a beam transport system, an ideal trajectory of an ideal particle is laid out. The ideal particle is one with the design energy or momentum and with a required initial trajectory (i.e., initially headed in the right direction). Magnetic elements are then arranged to guide this ideal particle to its final destination. However, particle accelerators and beam transport systems usually handle streams of many particles, typically billions at a time. Such beams will have a distribution of particles with an average energy that might be ideal but that has a spread in energy about this average. Likewise, not all particles will be headed along exactly the same trajectory but will have trajectories that start nearby. Therefore, beam transport systems need to control more than the "ideal" particle trajectory: they must appropriately control surrounding trajectories as well.
Static electric and magnetic fields, as shown above, can be used to control the trajectory of an ideal particle. However, a nearby particle with a slightly different trajectory needs to be guided back toward the ideal path. As a particle begins to deviate from the ideal course, one would like for it to be forced back toward its nominal position, much like what happens when a spring pushes and pulls a mass back toward its equilibrium location. Following this analogy, imagine a mass hanging motionless on a spring. When the mass is pulled and released, the spring exerts a force on the mass that is proportional to the displacement of the mass from its equilibrium point. The mass oscillates about the equilibrium point, undergoing simple harmonic motion.
A magnetic field whose strength is zero at one point and becomes stronger proportionally to the distance from the center is a quadrupole field. Such a field, derived from a quadrupole magnet, is depicted in Figure 2. In this figure, imagine a positively charged particle coming toward the reader. A particle moving down the center of the magnet will experience no force. However, particles displaced farther horizontally from the center will experience stronger forces directed back toward the center. One problem, however, is that particles that are displaced vertically from the center will experience forces that deflect them away from the center. This is a consequence of the fact that magnetic fields in free space are irrotational. Since one quadrupole field will focus the particle beam in one degree of freedom (horizontally, say) and defocus in the other degree of freedom (vertically, say), then one needs to examine arrangements of magnetic elements carefully to provide proper guiding (focusing) in both degrees of
freedom simultaneously. This topic is discussed further below.
Uniform magnetic fields used for guiding particle beams are typically generated by electromagnets with two poles (north and south) as depicted in Figure 3. In this traditional iron magnet, electric current is carried through the body of the magnet in copper conductors, and the lines of magnetic flux circulate around the copper, through the iron yoke, and into the magnet gap. The field in the gap, typically expressed in units of Tesla, is given by the equation where I is the current in the conductor, N is the number of conductor windings around each pole, d is the
height of the pole gap, and μ0 is the permeability of free space. The length of the magnet times the field generated inside the gap will determine the deflection of a particle's trajectory.
For focusing particle beams, a quadrupole magnet is typically used. A sketch of a quadrupole magnet design is provided in Figure 4. The vertical magnetic field in the gap is proportional to the horizontal displacement from the center, and the horizontal field is proportional to the vertical displacement from the center
In each case, the constant of proportionality, called the quadrupole gradient, is given by
where 2a is the distance between opposite pole tips. The gradient is typically expressed in units of Tesla/meter. To generate the desired quadrupole field, the iron poles are machined to an appropriate hyperbolic shape.
Likewise, the actual deflection of a particle trajectory due to a quadrupole field will depend on the length of the magnet. Since the deflection also depends on the displacement of the particle from the center of the quadrupole field, the magnet acts basically as a lens. If one considers the trajectory of a light ray passing through a simple lens, as depicted in Figure 5, one can see that the corresponding focal length of a quadrupole magnetic lens is given by
where p is the particle's momentum, q is its charge, L is the length of the magnet, and G is the quadrupole gradient defined above. As long as the length of the magnet is short compared to the focal length, the magnet can be treated as a "thin lens," and the focusing characteristics of the beam transport system can be understood by the standard rules of thin lens optics. One must be careful to note, however, that a lens which focuses in one degree of freedom will defocus in the other degree of freedom. Thus, vertical
and horizontal particle motion must be examined simultaneously in the magnetic optical system.
Also, it should be pointed out that other types of electromagnetic designs—for example, the coil configurations found in superconducting magnets—exist in addition to the simple examples cited above.
Beam Lines and Circular Accelerators
A beam transport system that delivers particles from one point to another is often referred to as a beam line. Such a system must transport the particles along the ideal trajectory, and the maximum displacement from the ideal trajectory of any given particle needs to lie within the physical aperture of the system. Any one focusing element along the way affects each degree of freedom differently; thus, the particle motion in each degree of freedom must be examined simultaneously. To analyze such a system's design, extreme initial conditions of possible particle trajectories can be traced through the system to be sure that they meet the final conditions required at the end of the beam line.
Tracking a particle's trajectory once around a circular accelerator, however, is not necessarily sufficient to determine the functionality of an accelerator's design. Circular accelerators take particles from one point back to the same point again, and again, and again. Thus, such a system must be "stable" under repetitive traversals. Further analysis of the basic magnetic system is therefore required.
Weak and Strong Focusing
Imagine a uniform magnetic field that is used to guide a particle in an ideal circular trajectory. If the field lines are pointed vertically, for instance, then horizontal motion is stable in this system. That is, if a particle begins its trajectory near but displaced slightly horizontally from the ideal circle, it will simply be guided around in a circular trajectory of the same radius as the ideal circle, but slightly offset. It will oscillate around the ideal trajectory with one oscillation per revolution. However, if the particle is given any vertical momentum, it will spiral around the vertical magnetic field lines and gain vertical displacement until it reaches the vacuum chamber walls; as shown in Figure 6.
Focusing in both the horizontal and vertical directions can be restored by widening the gap of the guiding magnet's pole face at its outside edge, as shown in Figure 7. The magnetic field lines are made to curve, and the magnetic field will become weaker near the radial outside and stronger near the radial center of the guiding magnets. Although this gives slightly less horizontal focusing, it provides vertical focusing. Since the force is always perpendicular to the field direction, a particle whose trajectory deviates in the vertical direction will now experience a force guiding it back toward the center of the magnet gap. So long as the wedge angle of the gap is not too steep, both horizontal and vertical motions will be stable and a particle will oscillate about the ideal circular trajectory. This form of focusing is called weak focusing and was commonly used in the design of particle accelerators until the mid-1950s.
Strong focusing can be understood by noting that a combination of two thin lenses, one focusing and one defocusing, separated by some appropriate distance will form a system that is itself focusing. In this way, a series of quadrupole magnets with their gradients alternating in sign (alternately focusing and defocusing in the horizontal direction, for instance) can create a system that focuses in both degrees of freedom simultaneously. As an example, motion through a repetitive system of equally spaced thin lens quadrupoles is stable as long as the lenses alternate between positive and negative focal lengths, and the absolute value of the focal length must be larger than half the distance between the lenses. This focusing structure, sometimes referred to as a FODO cell, is commonly used in long beam lines and large-circumference particle accelerators. The beam size can be kept arbitrarily small by focusing frequently enough and is not dependent on the overall length of the beam line or accelerator.
As the bending radii of weak focusing accelerators became larger and larger to accommodate higher and higher particle energies, their magnet apertures became larger accordingly and the amount of steel and copper required to generate the necessary magnetic fields made these devices very expensive to build. The invention of strong focusing in 1952 (Courant and Snyder) decoupled the focusing characteristics of the circular accelerator from its bend field requirements and thus allowed for accelerators of very large circumferences to be designed and built.
In an accelerator or beam line, dipole magnets, typically much smaller in strength than the main bending magnets, are used for the fine adjustment of the particle beam trajectory. By placing these steering magnets advantageously around the accelerator, the position and angle of the beam at important locations can be readily adjusted. Such uses might center the beam within a particle detector, as in a colliding beams experiment, or adjust the beam trajectory coming into the accelerator and onto a desired orbit. Independent control is often required over the particle beam position and its slope at a certain location. To perform a transverse position adjustment that is localized to the point in question, three steering magnets are necessary. With this so-called three bump system, the first magnet defines the trajectory leading to the desired position adjustment, and the second and third magnets bring the trajectory (position and slope) back to its original value.
While a three bump system can control a specific position in the beam line, independent control of both position and slope at a location requires two steering magnets upstream of the location in question and two downstream. Together, the two upstream magnets can be adjusted simultaneously to produce any desired position and slope, and then the two downstream magnets are adjusted to bring the trajectory back to its original state downstream of the system. Most large accelerators and beam lines are constructed with many such correctors to allow for adjustments to beam trajectories at arbitrary locations.
In addition to control of the beam trajectory, focusing adjustments are commonly required in beam transport systems as well. One example of their use would be to make fine adjustments to the number of oscillations a particle makes about the ideal trajectory in a circular accelerator. A small quadrupole magnet can be used to adjust the focusing characteristics of the horizontal oscillations, for example. However, the same magnet will also alter the vertical oscillations as well. Thus, two such quadrupole magnets are required for independent control of the oscillations in both degrees of freedom. For fine adjustments of the oscillation frequency, "families" of many small correction quadrupoles are typically located around the accelerator at favorable positions and connected in series electrically. Two independent families will control the horizontal and vertical motion independently. Having many such correctors reduces the necessary strength of the correction magnets and also serves to reduce the perturbations that the magnets make to the primary focusing structure of the system.
See also:Accelerator; Accelerators, Colliding Beam: Electron-Positron; Accelerators, Colliding Beam: Electron-Proton; Accelerators, Colliding Beam: Hadron; Accelerator, Fixed-Target: Electron; Accelerator: Fixed Target: Proton; Extraction System; Injector System
Chao, A. W., and Tigner, M., eds. Handbook of Accelerator Physics and Engineering (World Scientific, Singapore, 1999).
Conte, M., and MacKay, W. W. An Introduction to the Physics of Particle Accelerators (World Scientific, Singapore, 1991).
Courant, E. D., and Snyder, H. S. "Theory of the Alternating Gradient Synchrotron." Annals of Physics3 (1), 1–48(1958).
Edwards, D. A., and Syphers, M. J. An Introduction to the Physics of High Energy Accelerators (Wiley, New York, 1993).
Wilson, E. J. N. An Introduction to Particle Accelerators (Oxford University Press, Oxford, 2001).
Michael J. Syphers