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When a charged particle is accelerated in a uniform magnetic field it will move along a circular path in a plane perpendicular to the direction of the magnetic field. This is due to the fact that the magnetic force acting on a charged particle in a uniform magnetic field is always perpendicular to the velocity of the particle. So as the corresponding acceleration of the particle is always perpendicular to the magnetic field, the field doesn’t do any actual work on the particle, it just changes the direction of the velocity, as is shown in the diagram on the right.

The crosses are the tails of the magnetic field vectors, the circle is the path mapped out by the positive charge, the black arrows are the velocity vectors (v), and the green arrows are the magnetic force vectors (F). The red arrow represents the radius of the circle (r).

As the discoveries of the workings of the atom developed, physists wanted a way to accelerate protons to do experients on atomic nucli. One way of accelerating a proton would be to place it at one end of a long channel, and then apply a large negative voltage to the other, and use this to acclerate the positive proton towards it, and thus into the target nucleus. However, this way of doing things didn’t acclerate the protons fast enough to be of any real use, and dealing with the high voltages would have been both dangerous, expensive and complicated to say the least!

Then in 1932 Ernest Lawrence and Stanley Livingston developed the first 11-inch (27.5 cm) (classical) cyclotron magnetic resonance particle accelerator, based upon the principle above. It was capable of accelerating protons to energies of about 10 MeV without using a high voltage.

Nowadays though, modern particle accelerators can accelerate particles to speeds very close to the speed of light and take many forms and sizes, but are usually categorised into two main types, synchrotron (circular) and linac (linear), cyclotrons are the basis for the former, and the latter could be the cathode ray tube of a TV set, for a simple example.

The biggest and most complex cyclotron particle accelerator in the world is the TRIUMF accelerator in Canada, and accelerates protons to three quarters of the speed of light. The particles at TRIUMF make 1500 turns in 1/3000 of a second, with a total path length of 45 km. The accelerated protons can then also be used to form other sub-atomic particles such as
neutrons or other charged particles such as muons and pions (pions are very short lived mesons which are mass-produced in their billions every second at TRIUMF, using the very intense proton beams they have there) all travelling at high velocities.

The classical cyclotron basically consists of two hollow (copper) metal semi-circular electrodes called dees (due to their D shape), inside a vacuum chamber (to prevent collisions with air molecules), across which a high frequency oscillating voltage is put. The dees are open along their straight edge so that they effectively form a ‘broken cylinder’, and this is shown schematically in the diagrams on the right and on the next page.

Above and below the ‘broken cylinder’ there is a circular electromagnet, which provides a uniform magnetic field of fixed intensity across the dees and ‘acceleration cavity’. This is used to guide the travelling particles through the accelerator.

And in the very centre of the apparatus, between the dees there is an ion source, which releases the charged particles (protons) that are to be accelerated into the machine.

The source of the changing voltage is an oscillator (similar to that used for transmitting radio waves) that corresponds to the frequency of revolution of the particles in the magnetic field. This (accelerating) oscillating potential difference across the dees causes an electric field to be produced perpendicular to their plane, which is concentrated in the gap between them. The oscillation of electronic field in the cavity is then timed so that when the charged particle leaves one of the dees it is attracted towards the other.

It is only when the ions are in the gap between the dees that they are accelerated by the electric field, as when they enter the dees they feel no electric field (as no electric field exists within a conductor – Gauss’s law) but the ‘external’ magnetic field isn’t affected (shielded) by the metal of the dees and causes the linear path of the proton to curve around into a semi-circle and ’shoot’ back out of the other side of the dee, and into the gap again, where it is accelerated further.

The reason that the cyclotron actually works at all is the fact that the orbits of ions in a uniform magnetic field are isochronous; i.e. “the time taken by a particle of a given mass to make one complete circuit is the same at any speed or energy as long as the speed is much less than that of light”. This means that it is possible to accelerate the particles many times using a high voltage, which reverses its polarity at a constant frequency. This reversing of polarity at constant frequency has to fulfil the cyclotron’s ‘resonance condition’. In other words the ‘extra’ energy from the accelerating voltage must be ‘given’ to the proton at a frequency equal to the natural frequency of the proton circulating in the magnetic field for the proton to accelerate in a well-defined and predictable way. The equation for the resonance condition is qB = 2pmf_osc, and as m and q are fixed for the proton (and f_osc is assumed to be constant), we can see that we only need to alter the magnetic field, B until this condition is satisfied and a beam of energetic protons emerges.

As the particles move faster the radius of the path increases until they are ‘ejected’ from the accelerator at the required velocity, which can be found using the expression:

                        (1)

This equation can be adapted as follows to produce an expression fore the cyclotron frequency:

                        (2)

                        (3)

Equation (3) is characteristic of a particular particle moving in a particular magnetic field. It also tells us that the particles take exactly the same amount of time to travel along the larger semi-circular paths as they do for the initial shorter paths, which (again) is the critical characteristic in the operation of the cyclotron.

The frequency of the oscillator must match that of the cyclotron frequency, which, as is shown by equation (3) can be determined by the mass and charge of the particle, along with the value of the magnetic field. Again the final velocity of the particle can be found via equation (1) and for speeds much less than the speed of light the final kinetic energy can be given by K = ½ mv²

Classical cyclotrons have maximum energies of around 10 MeV. However, for particles of the same mass, but of higher charge the energies increase as the square of the charge.

Note that in the case of the cyclotron the final energy actually depends upon the magnetic field, and not the potential difference. The magnetic field can’t accelerate the particles, but it does control the number of revolutions that they make in the accelerator before they are ejected, and as the particles pick up energy on each revolution (i.e. on each crossing of the gap between the two dees) the more revolutions the particles make the greater their final energy.

If we increase the potential difference across the gap, the energy of the particle isn’t really affected as although the ‘thrust’ from the potential difference causes the particle to initially have more energy, it consequently moves in a path with a greater radius, the greater the radius the sooner the particle reaches the maximum radius where it leaves the accelerator. Therefore the particle crosses the gap fewer times and the net result is not that different from what would be achieved at lower potential difference, with the same magnetic field. And conversely, if the potential difference is lowered across the gap the particle gets less of a thrust on each revolution, but it makes more revolutions in total before it leaves, which again leads to the same resultant energy as a particle at higher potential difference for the same magnetic field.

As the speed of a particle approaches that of light, the effect of relativity causes its mass to increase. So the principles of the classical cyclotron have to be modified to allow for this relativistic increase. That is what happens in modern synchrocyclotrons, where speeds very close to that of light are achieved, with immense energies in the region of 1000 GeV, as at the Fermi-lab particle accelerator near Chicago – but that is something fairly different!

Conclusion:

Cyclotrons are essential for performing modern atomic and nuclear physics experiments where the building blocks of nature are constantly being probed and new particles are being found out. Without them, there would be great difficulties and expenses involved in accelerating the particles to high enough energies to perform even the most basic of experiments and essentially, if they didn’t exist we may not yet know about such things as quarks and mesons which are the foundations of what were previously thought to be some of the smallest particles imaginable (i.e. protons and neutrons). And really the best thing about the cyclotron in general, is that it’s a ready application of some very basic physical principles (neglecting the effect of relativity in synchrocyclotrons of course! – which in-itself is added evidence for relativity being real and not just a possibility).

Cyclotrons also have applications outside of academic research labs, for example, some of the applied research at TRIUMF involves, the use of a proton beam to destroy cancerous eye tumours, the development of a new method for detecting illegal drugs or plastic explosives hidden in mail, luggage and cargo containers, and they are also readily used in hospitals to create some of the short lived

radioactive particles used in medicine (as tracers for example).