Many small kicks instead of one large one
To study nuclei, physicists need beams of protons and other ions with energies of millions of electronvolts. Accelerating a proton to \(10\,\text{MeV}\) in one step needs a potential difference of ten million volts, which is very hard to hold. A cyclotron reaches the same energy with a modest voltage, by applying it again and again to the same ion. A magnetic field brings the ion back to the voltage every half turn.
The machine has two hollow, flat, D-shaped metal boxes, the dees, with a narrow gap between their straight edges. They lie between the poles of a large electromagnet, whose field crosses them at right angles. An oscillator connects the two dees, so they are alternately positive and negative. Ions start from a source at the centre of the gap.
Follow one proton. In the gap, the electric field pushes it into the dee that is negative at that moment, and it gains energy \(qV\). Inside the dee the metal walls shield it from the electric field, as the chapter Conductors in Electrostatics explains. Only the magnetic force acts there, so the proton turns a half circle at constant speed. It comes back to the gap moving the other way.
If the oscillator has reversed the dees by then, the proton is pushed across the gap again and gains another \(qV\). Each kick raises the speed, so the next half circle, \(r = mv/qB\), is larger. The path is a growing spiral of half circles. At the outer edge a deflector plate steers the beam out onto its target.
Staying in step
The whole design depends on timing. The oscillator must reverse the dees exactly as the ion finishes each half circle, however large the circle has grown. A half circle takes \(T/2 = \pi m/qB\), and that time does not depend on the speed or the radius. So one fixed frequency serves every turn. This is the resonance condition: the oscillator frequency must equal the ion's frequency of revolution.
Protons are accelerated in a cyclotron whose field is \(1.5\,\text{T}\). Find the oscillator frequency and the time for each half circle.
\(f = \dfrac{qB}{2\pi m} = \dfrac{1.6\times10^{-19}\times1.5}{2\pi\times1.67\times10^{-27}} = 2.29\times10^7\,\text{Hz}\), about \(23\,\text{MHz}\), in the radio band.
Each half circle takes \(\dfrac{1}{2f} = 2.19\times10^{-8}\,\text{s}\), about \(22\,\text{ns}\), at every radius.
Values used: \(e = 1.6\times10^{-19}\,\text{C}\), \(m_p = 1.67\times10^{-27}\,\text{kg}\), \(1\,\text{MeV} = 1.6\times10^{-13}\,\text{J}\).
Watch the gap in Fig 5.2. While the proton is inside a dee, the signs change over, and the field in the gap turns round. It has turned round exactly when the proton arrives. The half circles grow, from \(17.7\,\text{cm}\) after the first kick to \(50\,\text{cm}\) after the eighth, yet each one takes the same \(22\,\text{ns}\). After \(k\) equal kicks \(K\) grows as \(k\), so the radius grows as \(\sqrt k\).
Why can a cyclotron use an oscillator of one fixed frequency, even though its ions move faster and faster?
The cyclotron of the example (\(B = 1.5\,\text{T}\)) is used for deuterons, of charge \(e\) and mass \(2m_p\). What oscillator frequency is needed, in MHz?
The energy of the beam
The spiral ends at the edge of the dees, radius \(R\). There \(R = mv/qB\), so the ion leaves with \(v = qBR/m\). Its kinetic energy is \(\tfrac12 mv^2 = q^2B^2R^2/2m\). The gap voltage does not appear. A larger voltage gives larger kicks and fewer turns, but the final energy is set by the field and the size of the machine.
The proton cyclotron of the last example (\(B = 1.5\,\text{T}\)) has dees of radius \(0.5\,\text{m}\) and a gap voltage of amplitude \(50\,\text{kV}\). Find the energy of the beam in joules and in MeV, and the number of turns a proton makes.
The final energy comes from \(R\) and \(B\) alone; the gap voltage tells how much is gained on each turn.
\(K = \dfrac{q^2B^2R^2}{2m}\); energy per turn \(= 2qV_0\).
\(K = \dfrac{(1.6\times10^{-19})^2\times1.5^2\times0.5^2}{2\times1.67\times10^{-27}} = 4.31\times10^{-12}\,\text{J} = 26.9\,\text{MeV}\). Each turn gives \(2\times50\,\text{keV} = 100\,\text{keV}\), so the proton makes \(26.9/0.1 \approx 270\) turns.
The protons leave with about \(27\,\text{MeV}\), at \(7.2\times10^7\,\text{m/s}\), after about \(270\) turns.
The gap voltage of the cyclotron is doubled, and nothing else is changed. What happens?
The same machine (\(B = 1.5\,\text{T}\), \(R = 0.5\,\text{m}\)) is retuned for alpha particles, of charge \(2e\) and mass \(4m_p\). Find the energy of the alpha beam in MeV.
Where the cyclotron stops working
The resonance rests on the mass staying constant. At speeds close to the speed of light it does not: the unit on Modern Physics shows that a particle's mass grows with its speed. Then \(f = qB/2\pi m\) falls, and each half circle takes a little longer. The ion arrives at the gap later and later, until it meets the field pointing the wrong way. It stops gaining energy.
For protons this sets in slowly. At \(27\,\text{MeV}\) a proton's mass is only about \(3\,\%\) above its rest mass. An electron is about 1800 times lighter, and it nears the speed of light after less than a million volts. By \(0.5\,\text{MeV}\) its mass has nearly doubled. So an ordinary cyclotron is useless for electrons at high energy. Larger machines, the synchrocyclotron and the synchrotron, change the frequency or the field as the particles speed up.
Why is an ordinary cyclotron unsuitable for accelerating electrons to high energies?
A cyclotron works only if each half circle takes the same time, \(\pi m/qB\), so that a fixed oscillator stays in step. Electrons are so light that they reach speeds near the speed of light after a small energy gain. Their mass then grows with speed, so their frequency of revolution \(qB/2\pi m\) falls. They reach the gap later and later and fall out of step with the oscillator, and they stop gaining energy.
Ernest Lawrence, at the University of California in Berkeley, saw that the time for a half circle does not depend on the speed. In 1931 he and his student Stanley Livingston ran a cyclotron only a few inches across. It gave protons tens of thousands of electronvolts with a small voltage on its dees. Within a few years larger machines were splitting nuclei, and Lawrence received the Nobel Prize in 1939.
India's first large cyclotron was built at the Variable Energy Cyclotron Centre in Kolkata. Its first beam, of alpha particles, came out on 16 June 1977. The machine accelerates protons, deuterons, alpha particles and heavier ions, and its field and frequency are set to suit each one. Researchers from across the country use its beams for experiments in nuclear physics.
- Making the final energy depend on the gap voltage.\(K = q^2B^2R^2/2m\). The voltage sets only the number of turns.
- Counting one kick per turn.The ion crosses the gap twice in each turn, so it gains \(2qV_0\) per turn.
- Tuning the oscillator to half or twice the ion's frequency.The dees reverse twice per oscillation, once per half circle, so \(f_{osc}\) equals the frequency of revolution \(qB/2\pi m\).