A cyclotron is a type of particle accelerator invented by Ernest Lawrence in 1929–1930 at the University of California, Berkeley,^[1][2] and patented in 1932.^[3][4] A cyclotron accelerates charged particles outwards from the center of a flat cylindrical vacuum chamber along a spiral path.^[5][6] The particles are held to a spiral trajectory by a static magnetic field and accelerated by a rapidly varying electric field. Lawrence was awarded the 1939 Nobel Prize in Physics for this invention.^[6][7]

The cyclotron was the first "cyclical" accelerator.^[8] The primary accelerators before the development of the cyclotron were electrostatic accelerators, such as the Cockcroft–Walton generator and the Van de Graaff generator. In these accelerators, particles would cross an accelerating electric field only once. Thus, the energy gained by the particles was limited by the maximum electrical potential that could be achieved across the accelerating region. This potential was in turn limited by electrostatic breakdown to a few million volts. In a cyclotron, by contrast, the particles encounter the accelerating region many times by following a spiral path, so the output energy can be many times the energy gained in a single accelerating step.^[4]

Cyclotrons were the most powerful particle accelerator technology until the 1950s, when they were surpassed by the synchrotron.^[9] Nonetheless they are still widely used to produce particle beams for nuclear medicine and basic research. As of 2020, close to 1,500 cyclotrons were in use worldwide for the production of radionuclides for nuclear medicine.^[10] In addition, cyclotrons can be used for particle therapy, where particle beams are directly applied to patients.^[10]

History

In 1927, while a student at Kiel, German physicist Max Steenbeck was the first to formulate the concept of the cyclotron, but he was discouraged from pursuing the idea further.^[11] In late 1928 and early 1929, Hungarian physicist Leo Szilárd filed patent applications in Germany for the linear accelerator, cyclotron, and betatron.^[12] In these applications, Szilárd became the first person to discuss the resonance condition (what is now called the cyclotron frequency) for a circular accelerating apparatus. However, neither Steenbeck's ideas nor Szilard's patent applications were ever published and therefore did not contribute to the development of the cyclotron.^[13] Several months later, in the early summer of 1929, Ernest Lawrence independently conceived the cyclotron concept after reading a paper by Rolf Widerøe describing a drift tube accelerator.^[14][15][16] He published a paper in Science in 1930 (the first published description of the cyclotron concept), after a student of his built a crude model in April of that year. ^[17] He patented the device in 1932.^[4][18]

To construct the first such device, Lawrence used large electromagnets recycled from obsolete arc converters provided by the Federal Telegraph Company.^[19] He was assisted by a graduate student, M. Stanley Livingston. Their first working cyclotron became operational in January 1931. This machine had a diameter of 4.5 inches (11 cm), and accelerated protons to an energy up to 80 keV.^[20]

At the Radiation Laboratory on the campus of the University of California, Berkeley (now the Lawrence Berkeley National Laboratory), Lawrence and his collaborators went on to construct a series of cyclotrons which were the most powerful accelerators in the world at the time; a 27 in (69 cm) 4.8 MeV machine (1932), a 37 in (94 cm) 8 MeV machine (1937), and a 60 in (152 cm) 16 MeV machine (1939). Lawrence received the 1939 Nobel Prize in Physics for the invention and development of the cyclotron and for results obtained with it.^[21]

The first European cyclotron was constructed in the Soviet Union in the physics department of the V.G. Khlopin Radium Institute in Leningrad, headed by Vitaly Khlopin [ru]. This Leningrad instrument was first proposed in 1932 by George Gamow and Lev Mysovskii [ru] and was installed and became operative by 1937.^[22][23][24]

Two cyclotrons were built in Nazi Germany.^[25] The first was constructed in 1937, in Otto Hahn's laboratory at the Kaiser Wilhelm Institute in Berlin, and was also used by Rudolf Fleischmann. It was the first cyclotron with a Greinacher multiplier to increase the voltage to 2.8 MV and 3 mA current. A second cyclotron was built in Heidelberg under the supervision of Walther Bothe and Wolfgang Gentner, with support from the Heereswaffenamt, and became operative in 1943.^[26]

By the late 1930s it had become clear that there was a practical limit on the beam energy that could be achieved with the traditional cyclotron design, due to the effects of special relativity.^[27] As particles reach relativistic speeds, their effective mass increases, which causes the resonant frequency for a given magnetic field to change. To address this issue and reach higher beam energies using cyclotrons, two primary approaches were taken, synchrocyclotrons (which hold the magnetic field constant, but decrease the accelerating frequency) and isochronous cyclotrons (which hold the accelerating frequency constant, but alter the magnetic field).^[28]

Lawrence's team built one of the first synchrocyclotrons in 1946. This 184 in (4.7 m) machine eventually achieved a maximum beam energy of 350 MeV for protons. However, synchrocyclotrons suffer from low beam intensities (< 1 µA), and must be operated in a "pulsed" mode, further decreasing the available total beam. As such, they were quickly overtaken in popularity by isochronous cyclotrons.^[28]

The first isochronous cyclotron (other than classified prototypes) was built by F. Heyn and K.T. Khoe in Delft, the Netherlands, in 1956.^[29] Early isochronous cyclotrons were limited to energies of ~50 MeV per nucleon, but as manufacturing and design techniques gradually improved, the construction of "spiral-sector" cyclotrons allowed the acceleration and control of more powerful beams. Later developments included the use of more powerful superconducting magnets and the separation of the magnets into discrete sectors, as opposed to a single large magnet.^[28]

Principle of operation

Cyclotron principle

In a particle accelerator, charged particles are accelerated by applying an electric field across a gap. The force on a particle crossing this gap is given by the Lorentz force law:

where q is the charge on the particle, E is the electric field, v is the particle velocity, and B is the magnetic flux density. It is not possible to accelerate particles using only a static magnetic field, as the magnetic force always acts perpendicularly to the direction of motion, and therefore can only change the direction of the particle, not the speed.^[30]

In practice, the magnitude of an unchanging electric field which can be applied across a gap is limited by the need to avoid electrostatic breakdown.^[31]: 21 As such, modern particle accelerators use alternating (radio frequency) electric fields for acceleration. Since an alternating field across a gap only provides an acceleration in the forward direction for a portion of its cycle, particles in RF accelerators travel in bunches, rather than a continuous stream. In a linear particle accelerator, in order for a bunch to "see" a forward voltage every time it crosses a gap, the gaps must be placed further and further apart, in order to compensate for the increasing speed of the particle.^[32]

A cyclotron, by contrast, uses a magnetic field to bend the particle trajectories into a spiral, thus allowing the same gap to be used many times to accelerate a single bunch. As the bunch spirals outward, the increasing distance between transits of the gap is exactly balanced by the increase in speed, so a bunch will reach the gap at the same point in the RF cycle every time.^[32]

The frequency at which a particle will orbit in a perpendicular magnetic field is known as the cyclotron frequency, and depends, in the non-relativistic case, solely on the charge and mass of the particle, and the strength of the magnetic field:

where f is the (linear) frequency, q is the charge of the particle, B is the magnitude of the magnetic field that is perpendicular to the plane in which the particle is travelling, and m is the particle mass. The property that the frequency is independent of particle velocity is what allows a single, fixed gap to be used to accelerate a particle travelling in a spiral.^[32]

Particle energy

Each time a particle crosses the accelerating gap in a cyclotron, it is given an accelerating force by the electric field across the gap, and the total particle energy gain can be calculated by multiplying the increase per crossing by the number of times the particle crosses the gap.^[33]

However, given the typically high number of revolutions, it is usually simpler to estimate the energy by combining the equation for frequency in circular motion:

with the cyclotron frequency equation to yield:

The kinetic energy for particles with speed v is therefore given by:

where r is the radius at which the energy is to be determined. The limit on the beam energy which can be produced by a given cyclotron thus depends on the maximum radius which can be reached by the magnetic field and the accelerating structures, and on the maximum strength of the magnetic field which can be achieved.^[8]

K-factor

In the nonrelativistic approximation, the maximum kinetic energy per atomic mass for a given cyclotron is given by:

where ${\displaystyle e}$ is the elementary charge, ${\displaystyle B}$ is the strength of the magnet, ${\displaystyle r_{\max }}$ is the maximum radius of the beam, ${\displaystyle m_{a}}$ is an atomic mass unit, ${\displaystyle Q}$ is the charge of the beam particles, and ${\displaystyle A}$ is the atomic mass of the beam particles. The value of K

is known as the "K-factor", and is used to characterize the maximum kinetic beam energy of protons (quoted in MeV). It represents the theoretical maximum energy of protons (with Q and A equal to 1) accelerated in a given machine.^[34]

Particle trajectory

While the trajectory followed by a particle in the cyclotron is conventionally referred to as a "spiral", it is more accurately described as a series of arcs of constant radius. The particle speed, and therefore orbital radius, only increases at the accelerating gaps. Away from those regions, the particle will orbit (to a first approximation) at a fixed radius.^[35]

Assuming a uniform energy gain per orbit (which is only valid in the non-relativistic case), the average orbit may be approximated by a simple spiral. If the energy gain per turn is given by ΔE, the particle energy after n turns will be:

Combining this with the non-relativistic equation for the kinetic energy of a particle in a cyclotron gives: This is the equation of a Fermat spiral.

Stability and focusing

As a particle bunch travels around a cyclotron, two effects tend to make its particles spread out. The first is simply the particles injected from the ion source having some initial spread of positions and velocities. This spread tends to get amplified over time, making the particles move away from the bunch center. The second is the mutual repulsion of the beam particles due to their electrostatic charges.^[36] Keeping the particles focused for acceleration requires confining the particles to the plane of acceleration (in-plane or "vertical"^[a] focusing), preventing them from moving inward or outward from their correct orbit ("horizontal"^[a] focusing), and keeping them synchronized with the accelerating RF field cycle (longitudinal focusing).^[35]

Transverse stability and focusing

The in-plane or "vertical"^[a] focusing is typically achieved by varying the magnetic field around the orbit, i.e. with azimuth. A cyclotron using this focusing method is thus called an azimuthally-varying field (AVF) cyclotron.^[37] The variation in field strength is provided by shaping the steel poles of the magnet into sectors^[35] which can have a shape reminiscent of a spiral and also have a larger area towards the outer edge of the cyclotron to improve the vertical focus of the particle beam.^[38] This solution for focusing the particle beam was proposed by L. H. Thomas in 1938^[37] and almost all modern cyclotrons use azimuthally-varying fields.^[39]

The "horizontal"^[a] focusing happens as a natural result of cyclotron motion. Since for identical particles travelling perpendicularly to a constant magnetic field the trajectory curvature radius is only a function of their speed, all particles with the same speed will travel in circular orbits of the same radius, and a particle with a slightly incorrect trajectory will simply travel in a circle with a slightly offset center. Relative to a particle with a centered orbit, such a particle will appear to undergo a horizontal oscillation relative to the centered particle. This oscillation is stable for particles with a small deviation from the reference energy.^[35]

Longitudinal stability

The instantaneous level of synchronization between a particle and the RF field is expressed by phase difference between the RF field and the particle. In the first harmonic mode (i.e. particles make one revolution per RF cycle) it is the difference between the instantaneous phase of the RF field and the instantaneous azimuth of the particle. Fastest acceleration is achieved when the phase difference equals 90° (modulo 360°).^{[35]: ch.2.1.3} Poor synchronization, i.e. phase difference far from this value, leads to the particle being accelerated slowly or even decelerated (outside of the 0–180° range).

As the time taken by a particle to complete an orbit depends only on particle's type, magnetic field (which may vary with the radius), and Lorentz factor (see § Relativistic considerations), cyclotrons have no longitudinal focusing mechanism which would keep the particles synchronized to the RF field. The phase difference, that the particle had at the moment of its injection into the cyclotron, is preserved throughout the acceleration process, but errors from imperfect match between the RF field frequency and the cyclotron frequency at a given radius accumulate on top of it.^{[35]: ch.2.1.3} Failure of the particle to be injected with phase difference within about ±20° from the optimum may make its acceleration too slow and its stay in the cyclotron too long. As a consequence, half-way through the process the phase difference escapes the 0–180° range, the acceleration turns into deceleration, and the particle fails to reach the target energy. Grouping of the particles into correctly synchronized bunches before their injection into the cyclotron thus greatly increases the injection efficiency.^[35]: ch.7

Relativistic considerations

In the non-relativistic approximation, the cyclotron frequency does not depend upon the particle's speed or the radius of the particle's orbit. As the beam spirals outward, the rotation frequency stays constant, and the beam continues to accelerate as it travels a greater distance in the same time period. In contrast to this approximation, as particles approach the speed of light, the cyclotron frequency decreases due to the change in relativistic mass. This change is proportional to the particle's Lorentz factor.^{[30]: 6–9}

The relativistic mass can be written as:

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