Introduction

Daniel FitzGreen

Cyclotron Tuning

Introduction

The cyclotron was one of the first types of particle accelerators, patented by Ernest Lawrence in 1934. A cyclotron is a clever combination of a strong magnetic field and a rapidly-oscillating electric field.

As we know from the Lorentz force ( $F = q * v \times B$ ) and from the $e/m$ experiment, a charged particle moving perpendicularly to a magnetic field will have its trajectory bent into a circle. The Lorentz force will be equal to the centripetal force ( $F = mv^2/r$ ). One can solve for ‘ $r$ ’, determine the circumference of the circular trajectory, and determine the frequency at which the charged particle completes its round trip:

(1) $\begin{equation*} f_{\text{cyc}} = \cfrac{eB}{2\pi m} \end{equation*}$

Amazing! The cyclotron frequency is only dependent on the charge-to-mass ratio of the particle ( $e/m$ ) and the magnitude of the perpendicular magnetic field!

The charged particles are given energy by an oscillating electric field across two D-shaped electrodes within the magnetic field, as in Figure 1. The electric field oscillates and changes polarity at exactly the cyclotron frequency such that the charged particles receive a ‘kick’ as they pass through the gap between the electrodes on one side of the apparatus, and the opposite kick at the other, thereby accelerating twice per round trip.

The ingenious aspect of this apparatus is that the time the charged particles spend in the electrodes is the same regardless of their speed (since the radius of their trajectory increases). So the electric field can

Figure 1: Schematic of a cyclotron (magnet coils not shown – they are on either side of the electrodes, like a sandwich. The depicted ion trajectory is just a representation and is not the result of any calculations or modelling.

oscillate at a set frequency and the charged particles, emanating from the center of the instrument, will traverse a spiral until they are at the maximum radius allowed by the magnetic field.

The cyclotron at McMaster uses protons, which sets the value of $q$ and $m$ . The electric field of this cyclotron oscillates at a fixed frequency of 27.2 MHz. The only variable left to control is the magnetic field.

The magnetic field is provided by a huge set of Helmholtz coils. Cyclotron operators adjust the current going into the coils to find the magnetic field that generates the largest proton signal at the target. As the operator increases the current in the coils, the proton signal will increase, hit a maximum, then decrease. The signal as a function of magnet current can be described by a Gaussian function:

(2) $\begin{equation*} A*e^{\frac{-(x-X)^2}{2c^2}}\end{equation*}$

,
where $A$ is the amplitude, $X$ is the location of the peak amplitude, and $c$ is the full width at half max.

Introduction

License

Share This Book