Physics · Electricity and magnetism · Grade 11

Charged-particle accelerator

How does a cyclotron accelerate a proton, and how does a mass spectrometer separate isotopes?

The situation

At the Institute of Nuclear Physics near Tashkent there is a U-150-II cyclotron: it accelerates protons to 18 MeV and is used to produce radioactive isotopes. Inside it a particle moves along a spiral in a vacuum chamber under a magnetic field, circling hundreds of times and gaining a little energy on every turn. At the Large Hadron Collider near Geneva, protons fly at almost the speed of light around a ring nearly 27 km long. Why can a particle not be accelerated indefinitely in a cyclotron, and why does high energy need a giant ring?

Topic

Charged-particle accelerators. Cyclotron and mass spectrometer: separating isotopes

Key ideas

Lorentz force
Cyclotron frequency
Relativistic mass limit

Formulas

F = q v B, r = \frac{m v}{q B} = \frac{p}{q B}

The Lorentz force acts as the centripetal force: the particle moves in a circle.

f = \frac{q B}{2 π m}

The cyclotron frequency does not depend on speed — the RF generator frequency must equal it (non-relativistic).

E_{m a x} = \frac{q^{2} B^{2} R^{2}}{2 m} \to E = \sqrt{{(q B R c)}^{2} + {(m c^{2})}^{2}} - m c^{2}

Energy at the dee radius $R$ : the classical and the relativistic formula.

x = 2 r = \frac{2}{B} \sqrt{\frac{2 m U}{q}}

Mass spectrometer: an ion accelerated through the voltage $U$ lands on the plate at a distance $x$ , $x \propto \sqrt{m}$ .

B = \frac{p}{q ρ}, Δ E = \frac{e^{2} γ^{4} β^{3}}{3 ε_{0} ρ}

Synchrotron: the ring radius $ρ$ is constant and $B$ rises with the momentum; $Δ E$ is the synchrotron radiation loss per turn.

Theory

The Lorentz force and a circular orbit

A charge moving in a magnetic field experiences the Lorentz force $F = q v B$ ; it is perpendicular to the velocity, so it does no work — it only turns the direction. In a uniform field the particle describes a circle of radius $r = m v / (q B)$ (in the scene the red arrow is the force and the blue one is the velocity). The period $T = 2 π m / (q B)$ does not depend on speed: a fast particle traces a bigger circle but covers it in the same time. The same force deflects the charge carriers in a metal plate — the Hall effect — and separates the ions and electrons of a plasma flowing through a magnetic field onto opposite electrodes — the MHD generator (in detail in the “Magnetic field” and “Plasma — the fourth state” experiments).

The cyclotron and its energy limit

A high-frequency (RF) voltage is applied across the gap between two half-cylinder dees. The particle crosses the gap every half-turn and gains energy $q U$ , provided the voltage has changed sign by then: the resonance condition is $f_{R F} = q B / (2 π m)$ . At radius $R$ the energy reaches $E = q^{2} B^{2} R^{2} / (2 m)$ . But as the speed rises, the relativistic mass $γ m$ grows and the revolution frequency falls to $f / γ$ : the particle starts to lag behind the RF phase (the “Phase” graph) and, once past $90 °$ , begins to decelerate. That is why in a classical cyclotron protons can be accelerated only to about 10–20 MeV. The solutions are to lower the RF frequency along with the particle — the synchrocyclotron — or to raise $B$ with radius by a factor of $γ$ — the isochronous cyclotron (switch it on and try).

The mass spectrometer: separating isotopes

In the ion source the atoms are ionised, the voltage $U$ accelerates them to the speed $v = \sqrt{2 q U / m}$ , and then a magnetic field turns them through a semicircle. The landing point on the plate is $x = 2 r \propto \sqrt{m}$ : the heavier isotope lands farther away. The two isotopes of neon, with masses 20 and 22, were found in 1913 by J. J. Thomson by deflecting ion beams, and F. Aston confirmed this precisely in 1919 with a mass spectrograph he built himself. For $^{235}$ U and $^{238}$ U the mass difference is 1.3 %, and the relative difference between the landing points is only $\approx 0.6$ % — the beam width and the accuracy of the field are very important (see the task). Some spectrometers have a velocity filter first ( $E ⟂ B$ , $v = E / B$ ). Model: the ions are singly ionised and the beam width is $σ = 0.6$ mm.

The linear accelerator and the synchrotron

In a linear accelerator the particle passes through a row of drift tubes; there is no field inside a tube, while in the gaps between them an RF field accelerates it. For the particle to arrive at each gap in the right phase, the length of the $n$ -th tube must be $L_{n} = β_{n} λ / 2$ . An electron reaches $v \approx 0.94 c$ already at 1 MeV, so the tubes become almost identical. In a synchrotron the particle circulates in a ring of constant radius: as the momentum rises, the field of the bending magnets $B = p / (q ρ)$ is raised too, and the RF is matched to the revolution frequency. A ring in which two opposing beams are made to collide is a collider. LHC: circumference 26.7 km, 1 232 superconducting dipole magnets, proton energy up to 6.8 TeV.

Synchrotron radiation

A charge moving in a circle acquires centripetal acceleration and emits an electromagnetic wave. For a relativistic particle this radiation is gathered into a narrow cone along the direction of the velocity — synchrotron radiation. The loss per turn is $Δ E \propto γ^{4} / ρ$ : at the same energy $γ$ for an electron is 1 836 times greater than for a proton, and the loss is $1836^{4} \approx 10^{13}$ times greater. That is why electron rings are limited to an energy of $\sim 100$ GeV, and the highest energies are reached with protons. This radiation is also useful: special electron rings — synchrotron light sources — use X-rays to study the structure of proteins and materials.

Where do elementary particles come from?

Elementary particles are divided into leptons (the electron, muon, neutrinos), hadrons — baryons made of quarks (proton, neutron) and mesons (pions, kaons) — and the carriers of interactions, the bosons (photon, W, Z, gluon), plus the Higgs boson. Most are unstable, and they can be obtained from two sources: cosmic rays (high-energy protons and nuclei arriving at the atmosphere; the muons they produce reach the Earth's surface) and accelerators, in which particles are made to collide under controlled conditions. The positron and the muon were found in cosmic rays, and the Higgs boson was discovered at the LHC in 2012.

Did you know?

The first cyclotron was built by Ernest Lawrence in the USA in the early 1930s; for this invention he received the Nobel Prize in Physics in 1939.
In the LHC ring protons circle 11 245 times a second, and its 1 232 bending magnets work in the superconducting state at a temperature close to $- 271$ °C.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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