A cyclotron is a particle accelerator that uses a magnetic field and an alternating electric voltage to speed up charged particles. It is important because it can give ions high kinetic energy in a relatively compact circular machine. Cyclotrons are used in nuclear physics research, medical isotope production, and some cancer treatments.
The basic idea is to make a particle cross the same accelerating gap many times instead of using one very long accelerator.
Inside the cyclotron, two hollow D-shaped metal electrodes called dees sit in a uniform magnetic field. The magnetic field bends the moving charged particle into a circular path, while the electric field in the gap between the dees accelerates it each time it crosses. As the particle gains speed, its circular path grows into an outward spiral.
For nonrelativistic particles, the cyclotron frequency stays nearly constant, so the alternating voltage can be timed to push the particle forward on every crossing.
Understanding Physics: The Cyclotron
A particle begins near the centre from an ion source. The source removes or adds electrons so that an atom has a net charge. The chamber is pumped to a very low pressure.
This matters because collisions with air molecules would scatter the particle and drain its energy. Within each metal dee, the electric field is almost zero because the metal shields its interior. The particle therefore travels through a dee at nearly constant speed while the magnetic field continuously turns its path.
A magnetic field changes the direction of motion, not the particle's kinetic energy. The energy gain happens only in the narrow gap.
The timing of the voltage is the central challenge. Each time the particle reaches the gap, the voltage must have the polarity that pulls or pushes it forward. During the next half circle, the voltage reverses.
By the time the particle returns to the gap, it receives another forward push. This timing is called staying in phase. It works best for particles with a particular charge to mass ratio.
A proton, deuteron, or alpha particle needs a different operating frequency. If a particle arrives at the wrong time, the electric field can slow it down instead of speeding it up.
The orbit becomes wider because a faster particle needs a larger circle in the same magnetic field. At first, a standard cyclotron can keep the voltage rhythm matched to the orbit very well. At extremely high speeds, however, relativity becomes important.
The particle's momentum rises more quickly than the simple low speed model predicts. Its turning motion then takes slightly longer, so it gradually falls out of phase with a fixed alternating voltage.
A synchrocyclotron solves this by changing the voltage frequency over time. An isochronous cyclotron uses carefully shaped magnetic fields to help particles remain in step.
At the outer edge, the beam must be removed without sending it into the metal walls. Thin electric plates or magnetic devices nudge the particles onto an exit path. The beam can then strike a target.
In medical isotope production, the target material changes through nuclear reactions and forms short lived radioactive atoms for scans. In proton treatment systems, the beam energy is chosen so protons deposit much of their energy near a planned depth in tissue. This allows doctors to limit damage beyond the treatment region.
When learning cyclotrons, separate the jobs of the fields. The electric field transfers energy across the gap. The magnetic field bends the path inside the dees.
Pay close attention to direction. The force direction depends on whether the particle is positive or negative, as well as the direction of its motion and the magnetic field. A right hand rule can help with positive charges, while a negative charge feels a force in the opposite direction.
Units are useful checks too. Stronger magnetic fields make tighter paths, while heavier particles need larger paths at the same speed.
Key Facts
- Magnetic force on a moving charge: F = qvB when v is perpendicular to B.
- Centripetal force in the cyclotron: qvB = mv^2/r.
- Orbit radius: r = mv/(qB).
- Cyclotron angular frequency: omega = qB/m.
- Cyclotron frequency: f = qB/(2 pi m).
- Kinetic energy after acceleration: K = 1/2 mv^2, so larger radius means larger speed and energy.
Vocabulary
- Cyclotron
- A device that accelerates charged particles in a spiral path using a magnetic field and an alternating electric voltage.
- Dee electrode
- A hollow D-shaped metal electrode in a cyclotron where particles move in curved paths between acceleration gaps.
- Accelerating gap
- The narrow space between the dees where an electric field speeds up the charged particle.
- Ion source
- The central part of a cyclotron that produces the charged particles to be accelerated.
- Cyclotron frequency
- The frequency at which a charged particle circles in a uniform magnetic field when relativistic effects are small.
Common Mistakes to Avoid
- Using the electric field to explain the circular motion, which is wrong because the magnetic force provides the centripetal force inside the dees.
- Forgetting that acceleration happens mainly in the gap, which is wrong because the electric field is shielded inside the metal dees.
- Thinking the particle moves in a perfect circle forever, which is wrong because its speed and radius increase after each gap crossing.
- Using f = qB/m instead of f = qB/(2 pi m), which is wrong because qB/m is the angular frequency in radians per second.
Practice Questions
- 1 A proton moves in a cyclotron with magnetic field B = 0.80 T. Using q = 1.60 x 10^-19 C and m = 1.67 x 10^-27 kg, calculate its cyclotron frequency f = qB/(2 pi m).
- 2 An ion of mass 6.64 x 10^-27 kg and charge 3.20 x 10^-19 C moves at 2.0 x 10^6 m/s in a 1.5 T magnetic field. Calculate the radius of its path using r = mv/(qB).
- 3 Explain why the alternating voltage must reverse direction every half orbit for the particle to keep gaining energy.