Chapter 8. Particle Accelerators

A device that provides forces on charged particles by some combination of electric and magnetic fields and brings the ions to high speed and kinetic energy is called an accelerator. Many types have been developed for the study of nuclear reactions and basic nuclear structure, with an ever-increasing demand for higher particle energy. In this chapter we will review the nature of the forces on charges and describe the arrangement and principle of operation of several important kinds of particle accelerators. In later chapters we describe some of the many applications.

8.1. Electric and Magnetic Forces

Let us recall how charged particles are influenced by electric and magnetic fields. First, visualize a pair of parallel metal plates separated by a distance d as in the sample capacitor shown in Figure 8.1. A potential difference V and electric field ϵ = V/d are provided to the region of low gas pressure by a direct-current voltage supply such as a battery. If an electron of mass m and charge e is released at the negative plate, it will experience a force ϵe, and its acceleration will be ϵe/m. It will gain speed, and on reaching the positive plate it will have reached a kinetic energy . Thus its speed is . For example, if V is 100 volts, the speed of an electron (m = 9.1 × 10⁻³¹ kg and e = l.60 × 10⁻¹⁹ coulombs) is found to be 5.9 × 10⁶ m/s.

Figure 8.1. Capacitor as accelerator.

Next, let us introduce a charged particle of mass m, charge e, and speed υ into a region with uniform magnetic field B, as in Figure 8.2. If the charge enters in the direction of the field lines, it will not be affected, but if it enters perpendicularly to the field, it will move at constant speed on a circle. Its radius, called the radius of gyration, is r = mυ/eB, such that the stronger the field or the lower the speed, the smaller will be the radius of motion. Let the angular speed be ω (omega) equal to υ/r. By use of the formula for r, we find ω = eB/m. If the charge enters at some other angle, it will move in a path called a helix, like a wire door spring.

Figure 8.2. Electric charge motion in uniform magnetic field B.

Instead, let us release a charge in a region where the magnetic field B is changing with time. If the electron were inside the metal of a circular loop of wire of area A as in Figure 8.3, it would experience an electric force induced by the change in magnetic flux BA. The same effect would take place without the presence of the wire, of course. Finally, if the magnetic field varies with position, there are additional forces on charged particles.

Figure 8.3. Magnetic induction.

8.2. High-Voltage Machines

One way to accelerate ions to high speed is to provide a large potential difference between a source of charges and a target. In effect, the phenomenon of lightning, in which a discharge from charged clouds to the earth takes place, is produced in the laboratory. Two devices of this type are commonly used. The first is the voltage multiplier or Cockroft–Walton machine, Figure 8.4, which has a circuit that charges capacitors in parallel and discharges them in series. The second is the electrostatic generator or Van de Graaff accelerator, the principle of which is sketched in Figure 8.5. An insulated metal shell is raised to high potential by bringing it charge on a moving belt, permitting the acceleration of positive charges such as protons or deuterons. Particle energies of the order of 5 MeV are possible, with a very small spread in energy.

Figure 8.4. Cockroft–Walton circuit.

Figure 8.5. Van de Graaff accelerator.

8.3. Linear Accelerator

Rather than giving a charge one large acceleration with a high voltage, it can be brought to high speed by a succession of accelerations through relatively small potential differences, as in the linear accelerator (“LINAC”), sketched in Figure 8.6. It consists of a series of accelerating electrodes in the form of tubes with alternating electric potentials applied as shown. An electron or ion gains energy in the gaps between tubes and “drifts” without change of energy while inside the tube, where the field is nearly zero. By the time the charge reaches the next gap, the voltage is again correct for acceleration. Because the ion is gaining speed along the path down the row of tubes, their lengths ℓ must be successively longer for the time of flight in each to be constant. The time to go a distance ℓ is ℓ/υ, which is equal to the half-period of the voltage cycle T/2. The LINAC at the Stanford Linear Accelerator Center (SLAC) is 2 miles long. It produces electron and positron beams with energies up to 50 GeV (see References).

Figure 8.6. Simple linear accelerator.

8.4. Cyclotron and Betatron

Successive electrical acceleration by electrodes and circular motion within a magnetic field are combined in the cyclotron, invented by Ernest O. Lawrence. As sketched in Figure 8.7, ions such as protons, deuterons, or alpha particles are provided by a source at the center of a vacuum chamber located between the poles of a large electromagnet. Two hollow metal boxes called “dees” (in the shape of the letter D) are supplied with alternating voltages in correct frequency and opposite polarity. In the gap between dees, an ion gains energy as in the linear accelerator, then moves on a circle while inside the electric-field–free region, guided by the magnetic field. Each crossing of the gap with potential difference V gives impetus to the ion with an energy gain Ve, and the radius of motion increases according to r = υ/ω, where ω = eB/m is the angular speed. The unique feature of the cyclotron is that the time required for one complete revolution, T = 2π/ω, is independent of the radius of motion of the ion. Thus it is possible to use a synchronized alternating potential of constant frequency ν, angular frequency ω = 2πν, to provide acceleration at the right instant.

Figure 8.7. Cyclotron.

For example, in a magnetic field B of 0.5 Wb/m² (tesla) the angular speed for deuterons of mass 3.3 × 10⁻²⁷ kg and charge 1.6 × 10⁻¹⁹ coulombs is

Equating this to the angular frequency for the power supply, ω = 2πν, we find ν = (2.4 × 10⁷)/2π = 3.8 × 10⁶ s⁻¹, which is in the radiofrequency (RF) range.

The path of ions is approximately a spiral. When the outermost radius is reached and the ions have full energy, a beam is extracted from the dees by special electric and magnetic fields and allowed to strike a target, in which nuclear reactions take place.

The cyclotron is primitive compared with new machines but is still widely used in hospitals to produce radioisotopes.

In the betatron, the induction accelerator, electrons are brought to high speeds. A changing magnetic flux provides an electric field and a force on the charges while they are guided in a path of constant radius. Figure 8.8 shows the vacuum chamber in the form of a doughnut placed between specially shaped magnetic poles. The force on electrons of charge e is in the direction tangential to the orbit of radius r. The rate at which the average magnetic field within the loop changes is ΔB/Δt, provided by varying the current in the coils of the electromagnet. The magnitude of the force is^[†]

^† To show this, note that the area within the circular path is A = πr² and the magnetic flux is Φ = BA. According to Faraday's law of induction, if the flux changes by ΔΦ in a time Δt, a potential difference around a circuit of V = ΔΦ/Δt is produced. The corresponding electric field is ϵ = V/2πr, and the force is eϵ. Combining, the relation quoted is obtained.

Figure 8.8. Betatron.

The charge continues to gain energy while remaining at the same radius if the magnetic field at that location is half the average field within the loop. The acceleration to energies in the million-electron-volt range takes place in the fraction of a second that it takes for the alternating magnetic current to go through a quarter-cycle.

The speeds reached in a betatron are high enough to require the use of relativistic formulas (Chapter 1). Let us find the mass m and speed υ for an electron of kinetic energy E^k = 1 MeV. Rearranging the equation for kinetic energy, the ratio of m to the rest mass m⁰ is

Recalling that the rest energy E⁰ = m⁰c² for an electron is 0.51 MeV, we find the ratio m/m⁰ = 1 + 1/0.51 = 2.96. Solving Einstein's equation for the speed, , we find that . Thus the 1 MeV electron's speed is close to that of light, c = 3.0 × 10⁸ m /s (i.e., υ = 2.8 × 10⁸ m /s). If instead we impart a kinetic energy of 100 MeV to an electron, its mass increases by a factor 297 and its speed becomes 0.999995c.

Calculations of this type are readily made by use of the computer program ALBERT, introduced in Section 1.7. Some other applications to ion motion in modern accelerators are found in Computer Exercises 8.A and 8.B.

8.5. Synchrotron and Collider

Over the past half-century, the science and engineering of accelerators has evolved dramatically, with ever-increasing beam currents and energy of the charged particles. A major step was the invention independently of the synchrotron by E. M. McMillan and V. I. Veksler. It consists of the periodic acceleration of the particles by radiofrequency electric fields, but with a time-varying magnetic field that keeps the charges on a circular path. Ions that are out of step are brought back into step (i.e., they are synchronized). Figure 8.9 shows schematically the Cosmotron, operated from 1953 to 1966 at Brookhaven National Laboratory. An ion source provided protons that were injected at 4 MeV into a vacuum chamber by a Van de Graaff accelerator. The inflector sent the charges into the magnet. There, the magnetic field rose to 1.4 tesla in one second to provide the constant radius condition r = mυ /eB as the protons gain energy. The field was shaped to assure proper focusing. The radiofrequency unit accelerated the particles with initial voltage 2000 V at frequency 2000 hertz. Ions at final energy 3 GeV struck an internal target to yield neutrons or mesons.

Figure 8.9. Cosmotron. Synchrotron at Brookhaven National Laboratory.

In a more modern version of synchrotron, the magnetic field that bends the particles in a circular orbit is provided by a series of separate magnets, like beads on a necklace. In between the magnets are quadrupole (2N and 2S) magnets that provide beam focusing, helping compensate for space charge spreading.

Most of the early accelerators involved charge bombardment of a fixed target. Recently, much larger energies are achieved by causing two oppositely circulating beams to collide in what is called a storage ring. The pairs of particles used in a “collider” are (a) electrons and positrons, (b) protons and antiprotons, or (c) protons and protons. The accelerating cavity of the electron-positron collider at the Thomas Jefferson Accelerator Laboratory is constructed of superconducting niobium to minimize energy losses. It provides a total energy of 4 GeV. The Large Electron Positron (LEP) collider at the European Laboratory for Particle Physics (CERN) gave particles of 209 GeV before being shut down in 2000.

To reach high particle energies, a combination of accelerators of different types is used, as in the Tevatron at the Fermi National Accelerator Laboratory (Fermilab) near Chicago. The Tevatron involves a circular underground tunnel of diameter 3 m and length 6.3 km, containing the beam tube and a series of hundreds of magnets that provide ion bending. Negative hydrogen ions are first accelerated to 0.75 MeV by a Cockroft–Walton machine (Section 8.2) then raised to 200 MeV by a linear accelerator (Section 8.3). Electrons are stripped from the ions by a carbon foil, leaving protons. These are brought to 8 GeV by a small booster synchrotron. The ions are then injected into the Main Ring synchrotron and brought to 150 GeV. They are focused into short pulses and extracted to strike a copper target, creating large numbers of antiprotons. These are drawn off into a storage ring where they circulate and the beam is compressed, then transferred to an accumulator ring, and then put in the Tevatron ring. In the meanwhile a batch of protons from the Main Ring have also been put in the Tevatron ring. Along the path of that ring are 1000 superconducting magnets that use liquid nitrogen and helium for cooling. Finally, the two countercurrent beams, of diameter approximately 0.1 mm, are accelerated to their peak energy of nearly 1 TeV. Detection of the byproducts of collisions is by the Collider Detector Fermilab (CDF), a complex particle tracking device. Extensive additional information with photographs is found in the Fermilab Web site (see References).

Among the purposes of accelerators is the search for new particles in nature, which can be created only by transforming the energy of accelerated charges, in accord with Einstein's theory. Colliding high-energy beams of particles and antiparticles can create far more massive nuclear species than can simple ion bombardment of stationary targets. The reason is that a high-energy charge expends most of its energy in accelerating new particles to meet momentum conservation requirements. In contrast, when a particle collides with an antiparticle, the momentum is zero, allowing all of the energy to go into new mass.

One major accomplishment of high-energy machines was the discovery of the “top” quark (see References). Its existence is crucial to the correctness of the theory called the Standard Model. According to that picture, matter is composed of leptons (electrons, neutrinos, etc.) and quarks (types “up,” “down,” “charm,” “strange,” “top,” and “bottom”), along with their antiparticle forms. The up quark has a charge 2 /3, the down quark −1 /3. Quarks are believed to have been free just after the Big Bang, forming what is viewed as a perfect liquid. They clustered together with the help of gluons to form protons and neutrons. The proton is made of two ups and one down, whereas the neutron is two downs and one up. In the collision of protons and antiprotons, it is actually the component quarks that collide. It is believed that the top quark existed in nature only in the first 10⁻¹⁶ second from the Big Bang that started the universe. Quarks can be freed with difficulty in the laboratory by collisions of very high-energy gold atoms. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory has detected the products of quark combination. For a readily accessible discussion of quarks, go to Wikipedia.

Forces in nature are thought to be provided by the exchange of bosons, an example of which is the photon, for electromagnetic force. There are three other forces—weak (involved in radioactivity), strong (for binding in nuclei), and gravity. The electromagnetic and weak forces are viewed as different aspects of a more general “electroweak” force.

Studies of collisions of high-energy particles are intended to obtain information on the origin of mass, along with an answer why there is so much invisible mass (“dark matter”) in the universe, along with the cause of accelerated expansion of the universe (“dark energy”). Questions to be addressed are the mass of neutrinos, the scarcity of antimatter, and extra dimensions of space. Also sought is a hypothetical heavy particle called the Higgs boson, which is thought to relate the vacuum of space to the existence of particles.

In the early 1990s, the United States had started to build in Texas a large superconducting supercollider (SSC) to give a beam of 20 TeV, but the project was canceled by Congress because of excessive cost. With the demise of the SSC, a considerable part of high-energy particle research by United States physicists was shifted to CERN, the European Laboratory for Particle Physics (see References). The United States Department of Energy allocated funds to help construct the Large Hadron Collider (LHC) and the ATLAS detector (see References), which will analyze the products of proton–proton collisions. The LHC will make use of the existing 27 km circumference tunnel at the French–Swiss border. By use of superconducting magnets and advanced accelerator technology, it will be able to collide particles each of 7 TeV. Alternately, it will handle beams of heavy ions such as lead with total energy 1250 TeV.

Two extensions of particle accelerators have opened up new opportunities for research and industrial applications. The first is synchrotron radiation (SR), based on the fact that if an electric charge is given an acceleration, it radiates light. At each of the bending magnets of a synchrotron or storage ring, experimental beams of X-rays are available. The beams are very narrow, with an angle given by E⁰ /E^k, the ratio of rest energy and kinetic energy. An example of an SR facility is the National Synchrotron Light Source at Brookhaven National Laboratory (see References). The second is free electron laser (FEL), in which electrons are brought to high speed in a LINAC and injected into a tube with magnets along its length. These provide an alternating field that accelerates the electrons to radiate photons. The light is reflected back and forth by mirrors at the ends of the tube and interacts with the circulating electrons rather than with atoms as in a conventional laser. FELs can produce frequencies ranging from infrared to gamma rays. A Web site lists FELs around the world (see References).

8.6. Spallation

High-energy charged particles from an accelerator can disrupt nuclei of target materials. Experiments at California radiation laboratories showed that large neutron yields were achieved in targets bombarded by charged particles such as deuterons or protons of several hundred MeV energy. New dramatic nuclear reactions are involved. One is the stripping reaction, Figure 8.10(A), in which a deuteron is broken into a proton and a neutron by the impact on a target nucleus. Another is the process of spallation in which a nucleus is broken into pieces by an energetic projectile. Figure 8.10(B) shows how a cascade of nucleons is produced by spallation. A third is “evaporation” in which neutrons fly out of a nucleus with some 100 MeV of internal excitation energy, see Figure 8.10(C). The average energy of evaporation of neutrons is approximately 3 MeV. The excited nucleus may undergo fission, which releases neutrons, and further evaporation from the fission fragments can occur.

Figure 8.10. Nuclear reactions produced by very high energy charged particles.

It has been predicted that as many as 50 neutrons can be produced by a single high-energy (500 MeV) deuteron. The large supply of neutrons can be used for a number of purposes: (a) physics and chemistry research; (b) production of new nuclear fuel, beneficial radioisotopes, or weapons tritium; and (c) burn unwanted plutonium or certain radioactive waste isotopes. Some of these applications will be discussed in later sections.

At Oak Ridge National Laboratory the Spallation Neutron Source (SNS) was put into operation in 2006. Design and construction of the Department of Energy facility was a cooperative effort of six laboratories (Argonne, Brookhaven, Lawrence Berkeley, Los Alamos, Oak Ridge, and Jefferson). A large linear accelerator produces high-speed protons to bombard a liquid mercury target. The particle energy is 1 GeV; the beam power is 1.4 MW. Neutrons are moderated by water and liquid hydrogen, and a time-of-flight device selects neutrons of desired energy. The SNS will serve many hundreds of researchers in neutron science from the United States and abroad, facilitating a great variety of programs, as discussed in Chapter 18.

8.7. Summary

Charged particles such as electrons and ions of light elements are brought to high speed and energy by particle accelerators, which use electric and magnetic fields in various ways. In the high-voltage machines a beam of ions is accelerated directly through a large potential difference, produced by special voltage multiplier circuits or by carrying charge to a positive electrode; in the linear accelerator, ions are given successive accelerations in gaps between tubes lined up in a row; in the cyclotron, the ions are similarly accelerated but move in circular orbits because of the applied magnetic field; in the betatron, a changing magnetic field produces an electric field that accelerates electrons to relativistic speeds; in the synchrotron, both radiofrequency and time-varying magnetic field are used. High-energy nuclear physics research is carried out through the use of such accelerators. Through several spallation processes, high-energy charged particles can produce large numbers of neutrons which have a variety of applications.

8.8. Exercises

1. Calculate the potential difference required to accelerate an electron to speed 2 × 10⁵ m /s.
2. What is the proper frequency for a voltage supply to a linear accelerator if the speed of protons in a tube of 0.6 m length is 3 × 10⁶ m /s?
3. Find the time for one revolution of a deuteron in a uniform magnetic field of 1 Wb /m².
4. Develop a working formula for the final energy of cyclotron ions of mass m, charge q, exit radius R, in a magnetic field B. (Use nonrelativistic energy relations.)
5. What magnetic field strength (Wb /m²) is required to accelerate deuterons in a cyclotron of radius 2.5 m to energy 5 MeV?
6. Performance data on the Main Ring proton synchrotron of Fermilab at Batavia, Illinois (see References) were as follows:

   • • Diameter of ring 2 km
     • Protons per pulse 6 × 10¹²
     • Number of magnets 954
     • Initial proton energy 8 GeV
     • Final proton energy 400 GeV
     • Number of revolutions 200,000
   • Find the proton energy gain per revolution.
   • Find the speed of the protons at final energy by use of relativistic formulas of Sections 1.4 and 8.4 (or computer program ALBERT, see Chapter 1)
   • Calculate the magnetic field at the final speed of the protons.
7. What is the factor by which the mass is increased and what fraction of the speed of light do protons of 200 billion-electron-volts have?
8. Calculate the steady deuteron beam current and the electric power required in a 500-GeV accelerator that produces 4 kg per day of plutonium-239. Assume a conservative 25 neutrons per deuteron.
9. By use of the relativistic formulas from Section 1.4, show that for very large particle energies the fractional difference in speed from that of light, f = (c − υ) /c, is accurately approximated by f = (1 /2) (m⁰ /m)². Find f for 50 GeV electrons of rest energy 0.511 MeV.
10. The velocities of protons and antiprotons in the 2 km diameter Tevatron ring are practically the same as the velocity of light, 299792458 m /s. Find the time for particles of final energy 1 TeV to traverse the circumference. How much error is there in this approximation?
11. The synchrotron radiation loss in joules of a charge e with rest mass m⁰ moving in a circle of radius R is given by Cohen (see References) aswhere γ = E /m⁰c², with E = mc² and ϵ⁰ ≅ 8.8542 × 10⁻¹² F /m. (a) Find an approximate formula for ΔE in keV for an electron as a function of energy in GeV and R in meters, when the speed is very close to the speed of light. (b) How much lower than the radiation from an electron is that from a proton of the same radius and energy? (c) Find a formula for the power radiated from an electron moving in a circle with speed much less than the speed of light, in terms of the acceleration.

Computer Exercises

1. Verify with the computer program ALBERT (Chapter 1) that 1 TeV protons have a speed that seems to be the same as the velocity of light. Calculate the fractional difference between υ and c with the formula derived in Exercise 8.9. Explain the discrepancy.
2. The electron-positron collider at Hamburg, Germany, produces 23 TeV particles.
   • What is the ratio of the electron's total energy to its rest energy (0.510998910 MeV). Check the result with the computer program ALBERT (Chapter 1) by supplying a kinetic energy of 2.3D7 (a double precision number).
   • If 23 TeV electrons could be induced to travel around the earth (radius 6378 km), how far behind a light beam would they arrive? See Exercise 8.9 for a useful formula.

8.9 References

Sessler and Wilson, 2007 Andrew Sessler, Edmund Wilson, Engines of Discovery: A Century of Particle Accelerators 2007 World Scientific Publishing Co Singapore Accelerators for research, medicine, and industry

Early Particle Accelerators-Ernest Lawrence and the Cyclotron Early Particle Accelerators-Ernest Lawrence and the Cyclotron

http://www.aip.org/history/lawrence/epa.htm http://www.aip.org/history/lawrence/epa.htm

New systems, from the 1930s New systems, from the 1930s.

Humphries, 1986 Stanley Humphries Jr., Principles of Charged Particle Acceleration 1986 John Wiley & Sons New York

Scharf, 1997 Waldemar H. Scharf, Biomedical Particle Accelerators 1997 AIP Press New York

Shafroth and Austin, 1997 Stephen M. Shafroth, James C. Austin, Accelerator-Based Atomic Physics Techniques and Applications 1997 AIP Press Woodbury, NY

Stanford Linear Accelerator Center Stanford Linear Accelerator Center

http://www2.slac.stanford.edu/vvc http://www2.slac.stanford.edu/vvc

Select Virtual Visitor Center Select Virtual Visitor Center

Rees, October 1990 John R. Rees, The Stanford Linear Collider Scientific American October 199058-

Fermilab History and Archives Project Fermilab History and Archives Project

http://history.fnal.gov http://history.fnal.gov

Features Robert R. Wilson, first director Features Robert R. Wilson, first director.

Fermi National Accelerator Laboratory Fermi National Accelerator Laboratory

http://www.fnal.gov http://www.fnal.gov

Select About Fermilab /Virtual Tour Select About Fermilab /Virtual Tour.

The Discovery of the Top Quark The Discovery of the Top Quark

http://www.hep.uiuc.edu/home/tml/SciAmTop.pdf http://www.hep.uiuc.edu/home/tml/SciAmTop.pdf

Scientific American feature article by Tony M. Liss and Paul L. Tipton Scientific American feature article by Tony M. Liss and Paul L. Tipton.

Particle Accelerators Around the World Particle Accelerators Around the World

http://www-elsa.physik.uni-bonn.de/accelerator_list.html http://www-elsa.physik.uni-bonn.de/accelerator_list.html

Links to facilities sorted by location and by accelerator type Links to facilities sorted by location and by accelerator type.

European Laboratory for Particle Physics (CERN) European Laboratory for Particle Physics (CERN)

http://www.cern.ch http://www.cern.ch

Select Search /LHC Select Search /LHC.

The ATLAS Experiment The ATLAS Experiment

http://atlas.ch http://atlas.ch

Select Virtual Tour Select Virtual Tour

National Synchrotron Light Source National Synchrotron Light Source

http://www.nsls.bnl.gov http://www.nsls.bnl.gov

Select About the NSLS /Visit the NSLS /Take an Online Tour Select About the NSLS /Visit the NSLS /Take an Online Tour.

Cohen, 1995 E. Richard Cohen, The Physics Quick Reference Guide 1995 American Institute of Physics Woodbury, NY

WWW Virtual Library: Free Electron Laser WWW Virtual Library: Free Electron Laser

http://sbfel3.ucsb.edu/www/vl_fel.html http://sbfel3.ucsb.edu/www/vl_fel.html

Select Jefferson Lab or University of California at Santa Barbara Select Jefferson Lab or University of California at Santa Barbara.

Spallation Neutron Source Spallation Neutron Source

http://neutrons.ornl.gov http://neutrons.ornl.gov

Extensive information about features of the system and prospective applications Extensive information about features of the system and prospective applications.

T. E. Mason, et al., “The Spallation Neutron Source: A powerful tool for materials research.” T. E. Mason, et al., “The Spallation Neutron Source: A powerful tool for materials research.”

http://arxiv.org/abs/physics/0007068 http://arxiv.org/abs/physics/0007068

A frequently cited article describing the equipment used in research. Select Download PDF A frequently cited article describing the equipment used in research. Select Download PDF.