= Relativistic detuning of a cyclotron
{title2=$\omega(K)=|q|B/(\gamma m)$}
At larger <kinetic energy>, <special relativity> changes the relation between <momentum> and <velocity>. With $m$ the invariant <mass>, the <Lorentz factor> is $\gamma=1+K/(mc^2)$, and a <relativistic cyclotron orbit> in a fixed uniform <magnetic field> satisfies
$$
p=\gamma mv=|q|Br,\qquad
\omega(K)=\frac{|q|B}{\gamma m},\qquad
r(K)=\frac{\sqrt{K(K+2mc^2)}}{|q|Bc}.
$$
The <cyclotron frequency> decreases as the <kinetic energy> increases. If the drive stays at the classical value $\omega_0=|q|B/m$, the half-turn time is $\pi\gamma/\omega_0$. During it the drive advances through $\pi\gamma$ rather than $\pi$ radians. Its excess <phase> per half-turn is therefore
$$
\delta\phi=\omega_0\tau_{1/2}-\pi=\pi(\gamma-1)=\frac{\pi K}{mc^2},\qquad
\frac{\omega(K)-\omega_0}{\omega_0}=-\frac{K}{mc^2+K}.
$$
Even a small fractional <frequency> mismatch accumulates across many turns. A particle eventually reaches a weak or decelerating gap <phase>, so the classical fixed-field, fixed-frequency design has a practical <kinetic energy> limit. The limit depends on the kick size and acceptable accumulated <phase> error, rather than one universal cutoff <energy>. Changing drive <frequency> or shaping the <magnetic field> can restore synchronism; the argument does not forbid relativistic circular accelerators.
For the illustrative <proton> at $1.00\,\mathrm{MeV}$, $\gamma-1\simeq0.00106579$: the <frequency> is lower by about $0.1065\%$, and the next half-turn adds about $0.19184^\circ$ of drive <phase> excess. To see cumulative sensitivity, prescribe an ideal ramp $K_j=j\epsilon$ with $\epsilon=10\,\mathrm{keV}$ and sum the errors of the first $N$ coasts after kicks:
$$
\Phi_N=\sum_{j=1}^N\frac{\pi K_j}{mc^2}
=\frac{\pi\epsilon N(N+1)}{2mc^2}.
$$
This gives $\Phi_{100}\simeq9.688^\circ$ after the coast following kick $100$; the arrival at kick $100$ instead has $\Phi_{99}\simeq9.496^\circ$. The prescribed ramp diagnoses detuning but does not feed the resulting <phase> error back into the kicks. A self-consistent thin-gap model uses the <cyclotron gap energy gain> with $\cos\phi_n$, then advances the <phase> with the new <kinetic energy> during the following coast.
\Image[/cyclotron-energy-radius-and-phase-slip.png]
{title=Orbit radius and accumulating timing error for a chosen proton cyclotron model}
{description=Original calculations for B = 1 T and prescribed 10 keV kicks, not historical data. Left: classical and relativistic curvature radius. Middle: excess drive phase during one half-turn at the indicated kinetic energy. Right: accumulated excess after N coasts for K_j = j times 10 keV. At 1 MeV the single-coast error is small but its accumulated value is already nearly ten degrees. The prescribed ramp omits feedback of timing error on energy gain.}
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