Cyclotron

Illustration of a linear accelerator, showing the increasing separation between gaps

Diagram of cyclotron operation from Lawrence's 1934 patent. The "D" shaped electrodes (left) are enclosed in a flat vacuum chamber, which is installed in a narrow gap between the two poles of a large magnet (right).

Vacuum chamber of Lawrence 69 cm (27 in) 1932 cyclotron with cover removed, showing the dees. The 13,000 V RF accelerating potential at about 27 MHz is applied to the dees by the two feedlines visible at top right. The beam emerges from the dees and strikes the target in the chamber at bottom.

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:

$${\displaystyle \mathbf {F} =q[\mathbf {E} +(\mathbf {v} \times \mathbf {B} )]}$$

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:

$${\displaystyle f={\frac {qB}{2\pi m}}}$$

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]

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:

$${\displaystyle f={\frac {v}{2\pi r}}}$$

with the cyclotron frequency equation to yield:

$${\displaystyle v={\frac {qBr}{m}}}$$

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

$${\displaystyle E={\frac {1}{2}}mv^{2}={\frac {q^{2}B^{2}r^{2}}{2m}}}$$

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]

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

$${\displaystyle {\frac {T}{A}}={\frac {(eBr_{\max })^{2}}{2m_{a}}}\left({\frac {Q}{A}}\right)^{2}=K\left({\frac {Q}{A}}\right)^{2}}$$

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

$${\displaystyle K={\frac {(eBr_{\max })^{2}}{2m_{a}}}}$$

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

The trajectory followed by a particle in the cyclotron approximated with a Fermat's spiral

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]

Still, a simple spiral may be a useful approximation. Considering that the particle gains energy ΔE in each turn, its energy after n turns will be:

$${\displaystyle E(n)=n\Delta E}$$

Combining it with the equation for the kinetic energy of a particle in the cyclotron gives:

$${\displaystyle r(n)={{\sqrt {2m\Delta E}} \over qB}{\sqrt {n}}}$$

This is the equation of a Fermat spiral.

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"[lower-alpha 1] focusing), preventing them from moving inward or outward from their correct orbit ("horizontal"[lower-alpha 1] focusing), and keeping them synchronized with the accelerating RF field cycle (longitudinal focusing).[35]

The in-plane or "vertical"[lower-alpha 1] 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 core of the magnet into sectors.[35] 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.[38]

The "horizontal"[lower-alpha 1] 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]

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

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:

$${\displaystyle m={\frac {m_{0}}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}={\frac {m_{0}}{\sqrt {1-\beta ^{2}}}}=\gamma {m_{0}},}$$

where:

• ${\displaystyle m_{0}}$ is the particle rest mass,
• ${\displaystyle \beta ={\frac {v}{c}}}$ is the relative velocity, and
• ${\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}}$ is the Lorentz factor.[30]^: 6–9

Substituting this into the equations for cyclotron frequency and angular frequency gives:

$${\displaystyle {\begin{aligned}f&={\frac {qB}{2\pi \gamma m_{0}}}\\[6pt]\omega &={\frac {qB}{\gamma m_{0}}}\end{aligned}}}$$

The gyroradius for a particle moving in a static magnetic field is then given by:[30]^: 6–9

$${\displaystyle r={\frac {\gamma \beta m_{0}c}{qB}}={\frac {\gamma m_{0}v}{qB}}={\frac {m_{0}}{qB{\sqrt {v^{-2}-c^{-2}}}}}}$$

Expressing the speed in this equation in terms of frequency and radius

$${\displaystyle v=2\pi fr}$$

yields the connection between the magnetic field strength, frequency, and radius:

$${\displaystyle \left({\frac {1}{2\pi f}}\right)^{2}=\left({\frac {m_{0}}{qB}}\right)^{2}+\left({\frac {r}{c}}\right)^{2}}$$

Since ${\displaystyle \gamma }$ increases as the particle reaches relativistic velocities, acceleration of relativistic particles requires modification of the cyclotron to ensure the particle crosses the gap at the same point in each RF cycle. If the frequency of the accelerating electric field is varied while the magnetic field is held constant, this leads to the synchrocyclotron.[32]

In this type of cyclotron, the accelerating frequency is varied as a function of particle orbit radius such that:

$${\displaystyle f(r)={\frac {1}{2\pi {\sqrt {\left({\frac {m_{0}}{qB}}\right)^{2}+\left({\frac {r}{c}}\right)^{2}}}}}}$$

The decrease in accelerating frequency is tuned to match the increase in gamma for a constant magnetic field.[32]

In isochronous cyclotrons, the magnetic field strength B as a function of the radius r has the same shape as the Lorentz factor γ as a function of the speed v.

If instead the magnetic field is varied with radius while the frequency of the accelerating field is held constant, this leads to the isochronous cyclotron.[32]

$${\displaystyle B(r)={\frac {m_{0}}{q{\sqrt {\left({\frac {1}{2\pi f}}\right)^{2}-\left({\frac {r}{c}}\right)^{2}}}}}}$$

Keeping the frequency constant allows isochronous cyclotrons to operate in a continuous mode, which makes them capable of producing much greater beam current than synchrocyclotrons. On the other hand, as precise matching of the orbital frequency to the accelerating field frequency is the responsibility of the magnetic field variation with radius, the variation must be precisely tuned.

An approach which combines static magnetic fields (as in the synchrocyclotron) and alternating gradient focusing (as in a synchrotron) is the fixed-field alternating gradient accelerator (FFA). In an isochronous cyclotron, the magnetic field is shaped by using precisely machined steel magnet poles. This variation provides a focusing effect as the particles cross the edges of the poles. In an FFA, separate magnets with alternating directions are used to focus the beam using the principle of strong focusing. The field of the focusing and bending magnets in an FFA is not varied over time, so the beam chamber must still be wide enough to accommodate a changing beam radius within the field of the focusing magnets as the beam accelerates.[40]

There are a number of basic types of cyclotron:[41]

Classical cyclotron

The earliest and simplest cyclotron. Classical cyclotrons have uniform magnetic fields and a constant accelerating frequency. They are limited to nonrelativistic particle velocities (the output energy small compared to the particle's rest energy), and have no active focusing to keep the beam aligned in the plane of acceleration.[33]

Synchrocyclotron

The synchrocyclotron extended the energy of the cyclotron into the relativistic regime by decreasing the frequency of the accelerating field as the orbit of the particles increased to keep it synchronized with the particle revolution frequency. Because this requires pulsed operation, the integrated total beam current was low compared to the classical cyclotron. In terms of beam energy, these were the most powerful accelerators during the 1950s, before the development of the synchrotron.[28][9]

Isochronous cyclotron (isocyclotron)

These cyclotrons extend output energy into the relativistic regime by altering the magnetic field to compensate for the change in cyclotron frequency as the particles reached relativistic speed. They use specially shaped magnet pole pieces to create a nonuniform magnetic field stronger in peripheral regions. Most modern cyclotrons are of this type. The pole pieces can also be shaped to cause the beam to keep the particles focused in the acceleration plane as the orbit. This is known as "sector focusing" or "azimuthally-varying field focusing", and uses the principle of alternating-gradient focusing.[28]

Separated sector cyclotron

Separated sector cyclotrons are machines in which the magnet is in separate sections, separated by gaps without field[28].

Superconducting cyclotron

"Superconducting" in the cyclotron context refers to the type of magnet used to bend the particle orbits into a spiral. Superconducting magnets can produce substantially higher fields in the same area than normal conducting magnets, allowing for more compact, powerful machines. The first superconducting cyclotron was the K500 at the Michigan State University, which came online in 1981.[42]

The particles for cyclotron beams are produced in ion sources of various types.

Proton beams

The simplest type of cyclotron beam, proton beams are typically created by ionizing hydrogen gas.[43]

H⁻ beams

Accelerating negative hydrogen ions simplifies extracting the beam from the machine. At the radius corresponding to the desired beam energy, a metal foil is used to strip the electrons from the H⁻ ions, transforming them into positively charged H⁺ ions. The change in polarity causes the beam to be deflected in the opposite direction by the magnetic field, allowing the beam to be transported out of the machine.[44]

Heavy ion beams

Beams of particles heavier than hydrogen are referred to as heavy ion beams, and can range from deuterium nuclei (one proton and one neutron) up to uranium nuclei. The increase in energy required to accelerate heavier particles is balanced by stripping more electrons from the atom to increase the electric charge of the particles, thus increasing acceleration efficiency.[43]

To make use of the cyclotron beam, it must be directed to a target.[45]

Internal targets

The simplest way to strike a target with a cyclotron beam is to insert it directly into the path of the beam in the cyclotron. Internal targets have the disadvantage that they must be compact enough to fit within the cyclotron beam chamber, making them impractical for many medical and research uses.[46]

External targets

While extracting a beam from a cyclotron to impinge on an external target is more complicated than using an internal target, it allows for greater control of the placement and focus of the beam, and much more flexibility in the types of targets to which the beam can be directed.[46]

For several decades, cyclotrons were the best source of high-energy beams for nuclear physics experiments. With the advent of strong focusing synchrotrons, cyclotrons were supplanted as the accelerators capable of producing the highest energies.[32][9] However, due to their compactness, and therefore lower expense compared to high energy synchrotrons, cyclotrons are still used to create beams for research where the primary consideration is not achieving the maximum possible energy.[42] Cyclotron based nuclear physics experiments are used to measure basic properties of isotopes (particularly short lived radioactive isotopes) including half life, mass, interaction cross sections, and decay schemes.[47]

Cyclotron beams can be used to bombard other atoms to produce short-lived isotopes with a variety of medical uses, including medical imaging and radiotherapy.[48] Positron and gamma emitting isotopes, such as fluorine-18, carbon-11, and technetium-99m[49] are used for PET and SPECT imaging. While cyclotron produced radioisotopes are widely used for diagnostic purposes, therapeutic uses are still largely in development. Proposed isotopes include astatine-211, palladium-103, rhenium-186, and bromine-77, among others.[50]

The first suggestion that energetic protons could be an effective treatment method was made by Robert R. Wilson in a paper published in 1946[51] while he was involved in the design of the Harvard Cyclotron Laboratory.[52]

Beams from cyclotrons can be used in particle therapy to treat cancer. Ion beams from cyclotrons can be used, as in proton therapy, to penetrate the body and kill tumors by radiation damage, while minimizing damage to healthy tissue along their path.

As of 2020, there were approximately 80 facilities worldwide for radiotherapy using beams of protons and heavy ions, consisting of a mixture of cyclotrons and synchrotrons. Cyclotrons are primarily used for proton beams, while synchrotrons are used to produce heavier ions.[53]

• The 88-Inch Cyclotron at Lawrence Berkeley National Laboratory
• PSI Proton Accelerator – the highest beam current cyclotron in the world.
• RIKEN Nishina Center for Accelerator-based Science – Home of the highest energy cyclotron in the world
• Rutgers Cyclotron – Students at Rutgers University built a 30 cm (12 in) 1 MeV cyclotron as an undergraduate project, which is now used for a senior-level undergraduate and a graduate lab course.
• TRIUMF – the largest single-magnet cyclotron in the world.

• Ernest Lawrence's Cyclotron A history of cyclotron development at the Berkeley Radiation Laboratory, now Lawrence Berkeley National Laboratory
• National Superconducting Cyclotron Laboratory of the Michigan State University – Home of coupled K500 and K1200 superconducting cyclotrons; the K500, the first superconducting cyclotron, and the K1200, formerly the most powerful in the world.