The cyclotron

The selector-spectrometer measures velocities fixed from outside. To accelerate particles to much higher energies, the cyclotron is used instead, invented by Ernest Lawrence in 1929 (Nobel Prize 1939).

The idea exploits a surprising fact: in circular motion in a uniform $\vec{B}$, the orbital period does not depend on the speed. Indeed, from $T = 2\pi r/v$ and $r = mv/(qB)$:

$T = \frac{2\pi r}{v} = \frac{2\pi m}{qB} \qquad \ev{f_c = \frac{q\,B}{2\pi\,m}}$

The frequency $f_c$ is called the cyclotron frequency. It is fixed only by the particle ($m$, $q$) and the field $B$: whether the particle is slow (small circle) or fast (large circle), it takes the same time to complete one revolution. A faster particle traces a longer circle at exactly the greater speed needed to keep the period constant.

Key formula — Cyclotron

$f_c = \frac{qB}{2\pi m} \quad(\text{independent of } v)$ $K_\text{max} = \frac{q^2 B^2 R^2}{2m}$

The cyclotron consists of two hollow half-discs (the D’s, or “dees”) separated by a small gap. An alternating voltage of frequency $f_c$ is applied between the two D’s. A particle injected at the centre traces a semicircle in the first D, is accelerated crossing the gap (where the voltage has the right direction), traces a larger semicircle in the second D, and so on. Since the period is constant, synchronism with the voltage is maintained every turn; the radius grows on each half-turn, up to the edge where the particle is extracted.

Cyclotron: two hollow D’s separated by a gap with an alternating voltage at frequency $f_c$. The particle spirals outward gaining energy on each pass through the gap.

The maximum kinetic energy depends on the radius $R$ of the device: when $r = R$, the velocity is $v_\text{max} = qBR/m$ and hence

$K_\text{max} = \tfrac{1}{2}m v_\text{max}^2 = \frac{q^2 B^2 R^2}{2m}$

Example — A small cyclotron for nuclear medicine

A cyclotron with $R = 0{,}5\;\text{m}$ and $B = 1{,}5\;\text{T}$ accelerates protons ($m_p \approx 1{,}67\cdot 10^{-27}\;\text{kg}$, $q = e$): $K_\text{max} = \frac{(1{,}6\cdot 10^{-19})^2\,(1{,}5)^2\,(0{,}5)^2}{2\cdot 1{,}67\cdot 10^{-27}} \approx 4{,}3\cdot 10^{-12}\;\text{J} \approx 27\;\text{MeV}$ with $f_c = qB/(2\pi m) \approx 23\;\text{MHz}$. Energies of this order are enough to produce short-lived radioactive isotopes (for example ${}^{18}\text{F}$) used in hospital PET scans.

From the cyclotron to the synchrotron

At higher energies the velocity approaches $c$ and the relativistic mass $\gamma m$ grows: the cyclotron frequency is no longer constant and synchronism is lost. Two tricks are then used: varying $f$ dynamically with $\gamma$ (synchro-cyclotron) or also varying $B$ while keeping the radius constant (synchrotron). Synchrotrons such as the LHC at CERN, with $R = 4{,}3\;\text{km}$, accelerate protons to $7\;\text{TeV}$, a million times the energy of a small hospital cyclotron.

Summary

Selector $v = E/B$: lets through a single velocity. Spectrometer: $r \propto m/q$. Cyclotron: $f_c = qB/(2\pi m)$ does not depend on $v$, $K_\text{max} = q^2 B^2 R^2/(2m)$.

Links

Topics: Magnetismo Concepts: Forza di Lorentz · Moto circolare uniforme · Energia cinetica

Related exercises: Esercizio svolto — ciclotrone da un chilometro · Ciclotrone semplice · Frequenza di ciclotrone di un protone