cyclotron

A cyclotron is a form of particle accelerator, invented by Ernest Lawrence in the early 1930s, which is useful both for low-to-medium-energy subatomic physics experiments and for commercial isotope production. It is based on the behavior of charged particles under crossed electric and magnetic fields.

The basic situation in a cyclotron is of a charged particle moving in a plane, under a constant uniform magnetic field directed perpendicular to that plane. The particle will move in a circle, with angular frequency qB/m, where q is the charge of the particle, m is the mass of the particle, and B is the magnetic field strength. (see cyclotron frequency for details) This frequency is independent of velocity, which is very important. If the particle is accelerated by some other force, the radius of its path will increase to maintain the constant frequency. In a cyclotron, this acceleration is provided by an oscillating electric field between two semi-circular plates called 'dees'. If the frequency of the oscillating electric potential is an integer multiple of the cyclotron frequency qB/m, then the particle sees a potential difference of the same sign every time it crosses the gap between the dees. The net result is that the particle accelerates in an outward spiral until its radius exceeds the radius of the cyclotron, at which point it leaves tangentially.

The cyclotron can, in theory, accelerate any charged particle to a velocity qBr/m, where r is the radius of the cyclotron. In practice, it is used mainly to accelerate protons and light ions, although use of a cyclotron to accelerate heavy ions is not unheard of. Light particles are used because their charge-to-mass ratio is relatively large, so lower magnetic fields and/or smaller radii are necessary to acheive a desired energy.

The largest cyclotron in the world is found at TRIUMF, in Vancouver, British Columbia. It produces a 520 MeV proton beam at a relatively high current. A clever method is used to extract proton beams with various energies in the 60-520 MeV range. Instead of accelerating protons (H⁺ ions), the accelerator accelerates H^- ions, consisting of a proton and two electrons. Thin foils are placed inside the accelerator which strip the electrons from the H^- ions, converting them to bare protons. Since the charge changes sign, the proton begins to circulate in the opposite direction and leaves the accelerator via a beam pipe at the edge. This method can be used to produce three separate beams simultaneously, which the TRIUMF lab takes full advantage of.

There are several disadvantages to the cyclotron design. The main disadvantage is the relatively low practical limit on beam energies. A cyclotron requires a circular disc shape to operate and a uniform magnetic field throughout the entire disc. Generating such a field is subject to immense practical problems; the TRIUMF accelerator uses an unusual design with pinwheel-shaped poles. A related problem is the ability for other accelerator designs, mainly synchrotrons, to have vastly larger accelerator sizes due to only requiring an accelerator ring and not an accelerator disc. Synchrotron rings of diameters measured in kilometers have been built, but the TRIUMF cyclotron, largest in the world, isn't even 20 meters in diameter. Nevertheless, there is no more practical design for a small accelerator due to its simplicity of operation.

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Derivation of the equation for the cyclotron frequency: Say you have a wire of cross-sectional area A and length L that contains particles, each with charge q and drift velocity v. Let's suppose that the time it takes for one particle to travel the length of the wire is t, so that v = L/t => L = vt. If we say the number of particles in the wire is N and the number of particles per unit volume is n, then n = N/AL, as AL is the volume of the wire. Hence: n = N/Avt => nAvt = N.

Now, the total charge that has travelled a distance L is Q, so Q = Nq => N = Q/q. Therefore: nAvt = Q/q => nAqv = Q/t. Current is defined as being the charge that passes a point in a wire per unit time, so I = Q/t. Finally, then: nAqv = I.

This, ladies and gentlemen, is what we call a breakthrough.

For now (until somebody messages me telling me how to derive this, or does it themselves), I will pull an equation out of thin air! F = BIL, where F is the force on a current-carrying wire, B is the magnetic flux density and L is the length of the wire. But I = nAqv. So: F = BnAqvL. As before, AL is the volume of the wire, and the number of particles per unit volume is n. Hence, the number of particles N = nAL. So, the force on one particle (call it f for the moment) is F/N, where F/N = f = nAqvL/nAL => f = Bqv.

You have all just witnessed a miracle of science. I hope you liked it, because here comes another one.

First things first. That small f looks ridiculous, so let's call it big F. From now on, F will mean the force on one particle. Let's say you have an electron on the floor, in a magnetic field, where the field lines are going straight downwards. For the purposes of illustration, I shall give the electron a kick to the east (not to the right, because then in becomes down and it doesn't work). Using Fleming's Left Hand Rule (Thumb = Force, Index finger = Field direction, Middle finger = Direction of conventional current, i.e. opposite direction to motion of electron), we can figure out that the electron is going to have a force southwards on it, so it will accelerate southwards. In fact, my exquisitely rectangular fingers tell me that the force will always be at right angles to the current, and so also to the motion of the electron (in the opposite direction).

This means that our electron will always have a force perpendicular to its motion; this force will remain constant if B, q and v are constant. If this doesn't make you slap your thigh, read up about circular motion in the weightlessness node. That's right. A constant force perpendicular to its motion means a constant force towards the centre of a circle, so the resultant force on the particle is its centripetal force, which must be m(v^2)/r. Therefore Bqv = m(v^2)/r => Bq = mv/r.

This is where another bit of circular motion comes in: v = rw, so: Bq = mrw/r => Bq = mw. A couple of new equations: w = 2(pi)/T, T = 1/f. Hence: w = 2(pi)/(1/f) => w = 2(pi)f. Substituting: Bq = 2(pi)mf => f = Bq/2(pi)m, where f is the angular frequency. As this is for an electron going round in a full circle, I imagine that the potential difference between the "dees" needs to be reversed twice as often, so that the cyclotron frequency is Bq/(pi)m.