# The Cyclotron

The Cyclotron

Figure 29-15 is a top view of the region of a cyclotron in which the particles (protons, say) circulate. The two hollow D-shaped objects (open on their straight edges) are  made of sheet copper. These dees, as they are called are part of an electrical oscillator that alternates the electric potential difference across the gap between the dees. The electrical signs of the dues are alternated so that the electric field in the gap alternates in direction, first toward one die and then the other due, back and forth. The dues are immersed in a magnetic field (B = 1.5 T) whose direction is out of the plan e of  the page and that is set up by a large electromagnet. suppose that a proton, injected by source S at the center of the cyclotron in  initially moves toward a negatively charged due. It will accelerate toward this die and enter it. Once inside, it is shielded from electric fields by the copper walls of the die; that is, the electric field does not enter the die. The magnetic field, however, is not screened by the (nonmagnetic) copper die, so the proton moves
in a circular path whose radius, w  ich dep.nds on its speed, is given by Eq. 29-16(r = mv/qB). Let us assume that at the instant the proton emerges into the center gap from the first dee, the potential difference  between the dies is  reversed. Thus, the proton   gain faces a negatively charged dee and is again accelerated. This process continues,the circulating proton always being in step with the oscillations of the dee  potential, until the proton has spiraled out to the edge of the dee system. There a  deflector plate sends it out through a portal.The key to the operation of the cyclotron is that the frequency I at which the proton circulates in the field (and that does not depend on its speed) must be equal to the fixed frequency loss of the electrical oscillator, or .

I = lose (resonance condition

This resonance condition says that, if the energy of the circulating proton is to increase, energy must be fed to it ,at a frequency lose that is equal to the natural frequency Iat which the proton circulates in the magnetic field. Combining Eqs. 29-18 and 29-23 allows us to write the resonance condition as

qB = 27Tmlosc

For the proton, q and m are fixed. The oscillator (we assume) is designed to work at a single fixed frequency lose. We then “tune” the cyclotron by varying B until Eq. 29-24 is satisfied and then many protons circulate through the magnetic field, to emerge as a beam.

The Proton Synchrotron

At proton energies above 50 MeV, the conventional cyclotron begins to fail because one of the assumptions of its design – that the frequency of revolution of a charged particle circulating in a magnetic field is independent of the particle’s speed-is true only for speeds that are much less than the speed of light. At greater proton speeds (above about 10% of the speed of light), we must treat the problem relativistically. According to relativity theory, as the speed of a circulating proton approaches that of light, the proton’s frequency of revolution decreases steadily. Thus,
the protons get out of step with the cycle on’s oscillator- whose frequency remains fixed at/use-and eventually the energy of the circulating proton stops increasing. There is another problem. For a 500 GeV proton in a magnetic field of 1.5 T, the path radius is 1.1 km. The corresponding magnet for a conventional cyclotron of the proper size would be impossibly expensive, the area of its pole faces being   bout 4 X 106 m2.The proton synchrotron is designed to meet these two difficulties. The magnetic field B and the oscillator frequency lose, instead of having fixed values as in the conventional cyclotron, are made to vary with t me during the accelerating cycle. When this is done properly, (I) the frequency of the circulating protons remains in step with the oscillator at all times, and (2) the protons follow a circular-not a  spiral-path. Thus, the magnet need extend only along that circular path, not overcome 4 X 106 m2. The circular path, however, still must be large if high energies are to be achieved. The proton synchrotron at the Fermi National Accelerator Laboratory (Fermilab) in lIIinois has a circumference of 6.3 km and can produce protons
with energies of about I TeV (= 1012 eV).

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