Each experiment compares the model with an independent route: a Boris-pusher integration of the full Lorentz force, a Larmor-power integral, a tracked trajectory or a four-vector calculation. Tolerances are stated per row.
| Quantity | Predicted | Measured | Error |
|---|
A charge q with momentum p = γ m v in a magnetic field moves on a circle of radius r = p/(qB). Energy is gained only from the electric field of an RF gap; the magnet merely bends. Relativity enters through γ = E/(mc²).
Cyclotron. Both D-shaped electrodes ("dees") are driven at one fixed RF frequency. Gain per gap crossing is ΔE = q V cos φ with φ the RF phase at the crossing; the sign of the field flips each half turn. As the proton gains energy γ grows, the orbit slows down and φ drifts: once φ passes 90° the particle is decelerated. That is the relativistic limit of the classic machine (of order 10–30 MeV for protons). The synchrocyclotron lowers the RF frequency to follow f = qB/(2πγm) of the reference particle, and phase stability (autophasing) captures the bunch.
Linac. A particle passes N gaps, gaining q V cos φ at each. In π-mode the RF flips sign while the particle crosses a drift tube, so the tube length is Ln = βn λ / 2 with λ = c/f. The synchronous phase φs is negative (field rising): a late particle sees a larger field and catches up, an early one gets less. This is longitudinal phase stability.
Synchrotron. The orbit radius is held fixed by ramping the field with the momentum, and the RF frequency with the revolution frequency:
B = p / (q R), f_rev = β c / (2π R), f_RF = h f_rev turn map: E' = E + qV sin φ − U₀(E) φ' = φ + 2π h η (p − p_s)/p_s, η = 1/γ_t² − 1/γ² bucket: ½ k ΔE² + qV (cos φ + φ sin φ_s) = const, k = 2π h η/(β² E) stability: η cos φ_s < 0 (phase jump at transition γ = γ_t)The curve through the unstable fixed point is the separatrix; particles inside oscillate (synchrotron tune Qs), particles outside drift away. Radiation by a charge on a circle (Liénard) gives, per turn,
At equal energy a proton's γ is 1836 times smaller, so its U₀ is (me/mp)⁴ ≈ 10⁻¹³ of the electron's. That is why electron rings stop growing in energy where qV sin φs can no longer pay U₀ (the bucket collapses) and why the highest energy circular machines use protons.
Focusing. Alternating quadrupoles (FODO cell, length Lc, thin-lens focal length f, κ = Lc/2f) give cos μ = 1 − κ²/2, βmax,min = Lc(1 ± sin μ/2)/sin μ, beam size σ = √(ε β) with geometric emittance ε = εn/(βγ): it shrinks as the beam is accelerated (adiabatic damping). Unstable for κ ≥ 2.
Collider. Two equal counter-rotating beams: √s = 2E, whereas a beam of energy E hitting a fixed proton gives √s ≈ √(2 E m c²). Luminosity L = frev nb Nb² / (4π σ*²), σ* = √(ε β*).
Simplifications, stated honestly. Linear (thin-lens) optics, uniform dipole filling the ring (bending radius = R), no space charge, no beam-beam force, no chromaticity or nonlinear resonances, no injection/extraction kickers, longitudinal motion by a one-turn map rather than tracking inside the cavity, radiation treated as a smooth loss (a small noise term keeps electron beams from collapsing to zero size; no full quantum excitation), cyclotron has no vertical focusing model and the "ion" is a fully stripped carbon-12 nucleus. Beam size in the 3D view is exaggerated and the machine is not to scale. Scenes are generic, not a model of any specific laboratory, and not affiliated with any.
Real numbers for scale. Protons at 6.5 TeV have γ ≈ 6930 and travel at 99.999999 % of c (1 − β ≈ 1.04×10⁻⁸); two such beams give √s = 13 TeV. Isochronous and synchro-cyclotrons, drift-tube linacs and synchrotrons all appear in particle therapy, isotope production and research. References: E. M. McMillan (1945) and V. Veksler (1944) on phase stability; Courant and Snyder (1958) on strong focusing; J. D. Jackson, Classical Electrodynamics, ch. 14 (radiation); S. Y. Lee, Accelerator Physics; C. Boris (1970) pusher.
Pick a machine, press a preset, then raise the time warp. Fixed-frequency cyclotron: watch the energy peak and fall as the bunch slips out of phase; switch on frequency modulation and the bunch reaches ~160 MeV. Synchrotron: watch the bucket on the Graphs tab; try electrons with a high target energy and see the bucket close when U₀ approaches qV.