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How physics instruments measure

How a cyclotron keeps a proton on the accelerating beat

A cyclotron sends a proton through an accelerating gap again and again. A magnetic field brings it back; an electric field gives it another push. Those pushes accumulate only when the voltage points the useful way at each crossing. As the proton gains momentum, relativity changes its return time and the original RF clock begins to miss the beat.

Sandbox Physics ·

A useful push depends on the arrival time

Two hollow D-shaped electrodes face each other across a narrow gap. Inside either electrode, the electric field is approximately screened and the particle mainly bends. Across the gap, the voltage changes its kinetic energy. The voltage reverses between successive passages so that a proton travelling back the other way can receive another accelerating push.

The place reached in the electrical cycle when the particle arrives is its phase. Arrival near a useful voltage peak gives a strong push. Arrival after the field reverses can remove energy. A finite crossing also takes time: the voltage continues changing while the proton is in the gap, so the whole transit contributes to the work.

Sources: [1]

A bigger orbit eventually takes longer

At low speeds, doubling the speed approximately doubles the bending radius in a uniform field. Distance and speed grow together, leaving the time per turn almost unchanged. The same pair of electrodes can therefore accelerate the proton repeatedly.

At higher energies, momentum keeps increasing as speed approaches the speed of light. Momentum still determines the bending radius. The path grows faster than the speed, and each turn takes longer. Small delays accumulate until the particle arrives outside the accelerating half-cycle.

ωrev=qBγm\omega_{\mathrm{rev}}=\frac{qB}{\gamma m}
In a uniform field B, a particle with charge q and rest mass m revolves more slowly as its relativistic factor increases.

Sources: [2]

Change the clock or strengthen the outer field

A synchrocyclotron gradually lowers the RF frequency. The voltage waits longer for each return, according to a programme designed around a reference particle. A group of protons gains energy during that sweep. Their differing starting conditions still affect how closely they follow it.

An isochronous cyclotron instead raises the average field at larger radii, bending energetic particles faster to maintain nearly equal turn times. Real machines also vary the field around the circumference to keep the beam vertically focused. A prescribed radial curve illustrates synchronism in the central plane; transverse stability needs the full magnet design.

Sources: [2]

The last orbit must pass a physical exit

The outer turns crowd together. An extraction electrode must fit between them, with a thin septum separating the circulating beam from a deflecting channel. A proton entering on the useful side can leave towards an external beamline. One entering slightly differently may strike the septum. Turn separation and beam tails therefore affect extraction losses.

A collector belongs at the end of that path. Its signal comes from particles that reach its incoming face. Particles still circulating, or absorbed by a wall, contribute no collector event. Keeping the channel fixed lets different compensation programmes reveal their consequences through actual arrivals and losses.

Sources: [3]

Independent standards connect signal with energy

Consider an ideal collector that fully stops each arriving proton and gives a proportional electronic response. The digitizer returns a code. Independently known particle energies determine the code-to-energy relation; another known energy, excluded from the fit, checks it. Unknown collector codes can then be converted into a distribution.

An uncertain energy standard shifts many converted events together. More particles reduce sampling fluctuations, but that shared reference error remains. Separately tagged incoming particles establish detection efficiency. In a physical instrument, the stopping material and its response must also be measured. This connects an accelerated orbit, the extracted population and the final energy readout.

Sources: [4]

Sources and further reading

  1. CERN, Accelerating: radiofrequency cavities

    Reads the public explanation of timed RF electric acceleration; no LHC operating parameters are adopted.

  2. M. Seidel, Cyclotrons and Fixed Field Alternating Gradient Accelerators, arXiv:2105.04477v3

    Adopts printed pages 1–4 for frequency, rigidity and isochronicity, and 6–8 for phase slip and prescribed frequency/field compensation. Visually reads page 7, Fig. 3 and Eqs. 18–19. A given midplane profile is not a complete focusing magnet.

  3. M. Seidel, Injection and Extraction in Cyclotrons, CERN-2018-008-SP, 151–162

    Reads printed 151–153 and 157–159 on turn separation, deflection and septum extraction; visually inspects Fig. 7, page 158. PSI geometry, loss fractions and performance are not assigned to the teaching model.

  4. JCGM 100, Evaluation of measurement data, section 5

    Uses sensitivity coefficients in 5.1.3 and correlated-input propagation in 5.2 to explain energy-reference errors shared by multiple observations.

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