Cyclotron
Cyclotron
A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1934. It is used to accelerate charged particles, such as protons, to high energies. Cyclotrons are widely used in nuclear physics experiments, medical applications such as cancer radiotherapy, and the production of radioisotopes for medical diagnostics and research.
Principle of Operation
The cyclotron works on the principle that a charged particle moving perpendicular to a magnetic field experiences a centripetal force that causes it to move in a circular path. By applying a perpendicular magnetic field and an alternating electric field, the cyclotron can accelerate particles to high speeds in a spiral path.
The magnetic field is provided by a large electromagnet with two hollow "D"-shaped electrodes, called "dees", between its poles. An alternating voltage is applied to the dees, which accelerates the particles each time they cross the gap between them. As the particles gain energy, their paths spiral outward until they reach the edge of the dees and can be extracted for use.
Cyclotron Frequency
The frequency of the alternating voltage must match the cyclotron frequency of the particles, which is the frequency with which they orbit in the magnetic field. The cyclotron frequency ($f_c$) is given by:
$$ f_c = \frac{qB}{2\pi m} $$
where:
- $q$ is the charge of the particle,
- $B$ is the magnetic field strength,
- $m$ is the mass of the particle.
Key Formulas
The radius of the particle's path ($r$) at any point is given by:
$$ r = \frac{mv}{qB} $$
where $v$ is the velocity of the particle.
The kinetic energy ($KE$) of the particle when it exits the cyclotron is:
$$ KE = \frac{1}{2}mv^2 $$
Limitations
- Relativistic Effects: As particles approach the speed of light, their mass effectively increases due to relativistic effects, which changes the cyclotron frequency. This makes the cyclotron less effective for accelerating particles to very high speeds.
- Maximum Energy: The maximum energy that can be achieved is limited by the size of the dees and the strength of the magnetic field.
- Single Particle Type: A cyclotron is typically designed to accelerate only one type of particle, as the frequency of the electric field is fixed to match the cyclotron frequency of that particle.
Cyclotron vs. Synchrotron
| Feature | Cyclotron | Synchrotron |
|---|---|---|
| Principle | Fixed magnetic field and frequency | Variable magnetic field and frequency |
| Particle Energy | Limited by size and magnetic field | Can reach very high energies |
| Relativistic Effects | Not accounted for; limits max energy | Accounts for relativistic effects |
| Size | Compact | Large |
| Application | Medical, industrial, research | High-energy physics research |
Example
Suppose we want to calculate the kinetic energy of a proton accelerated by a cyclotron. The magnetic field strength is $1.5 \, \text{T}$, and the radius of the cyclotron is $0.5 \, \text{m}$.
First, we calculate the cyclotron frequency:
$$ f_c = \frac{qB}{2\pi m} = \frac{(1.602 \times 10^{-19} \, \text{C})(1.5 \, \text{T})}{2\pi (1.673 \times 10^{-27} \, \text{kg})} \approx 14.3 \, \text{MHz} $$
Next, we find the velocity of the proton when it exits the cyclotron:
$$ v = \frac{qBr}{m} = \frac{(1.602 \times 10^{-19} \, \text{C})(1.5 \, \text{T})(0.5 \, \text{m})}{1.673 \times 10^{-27} \, \text{kg}} \approx 7.2 \times 10^6 \, \text{m/s} $$
Finally, we calculate the kinetic energy:
$$ KE = \frac{1}{2}mv^2 = \frac{1}{2}(1.673 \times 10^{-27} \, \text{kg})(7.2 \times 10^6 \, \text{m/s})^2 \approx 4.3 \times 10^{-13} \, \text{J} $$
This example demonstrates how a cyclotron can be used to accelerate protons to high speeds, providing them with significant kinetic energy for various applications.