Particle Accelerator Cyclotron Frequency Calculator
Find the orbital (cyclotron) frequency, period, and radius of a charged particle circling in a uniform magnetic field, including the relativistic droop with energy.
🔄 What is the Particle Accelerator Cyclotron Frequency Calculator?
This cyclotron frequency calculator finds the orbital frequency, period, and radius of a charged particle circling inside a uniform magnetic field, the same physics that powers a cyclotron particle accelerator. Given a particle's rest energy, charge, the magnetic field strength, and its kinetic energy, it applies f_c = zeB/(2*pi*gamma*m) directly from SI constants and reports both the frequency and the underlying Lorentz factor and beta used to compute it.
Physicists and accelerator engineers use this calculation to understand cyclotron and synchrocyclotron design, to estimate the RF drive frequency needed for a given magnet and particle species, and to teach the classic isochronism property that made the earliest cyclotrons possible. It is also a standard teaching example for how special relativity limits a simple, elegant accelerator design.
A common point of confusion is why cyclotron frequency does not depend on speed at low energy but does at high energy. Non-relativistically, a faster particle has a proportionally larger orbit radius, and the two effects cancel exactly, keeping frequency constant. This isochronism only holds while the Lorentz factor gamma stays close to 1; once kinetic energy becomes a meaningful fraction of the particle's rest energy, gamma grows and the frequency droops, exactly the effect this calculator's chart makes visible.
This tool is intended as an educational and back-of-envelope design calculator, not a substitute for a full accelerator lattice simulation. It assumes a perfectly uniform field and ignores beam loading, space charge, and RF cavity design details that a real cyclotron engineer must also account for.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Proton at 10 MeV in a 1 T field
Example 2 - Electron at 1 MeV in a 0.5 T field
Example 3 - Alpha particle at 50 MeV in a 2 T field
❓ Frequently Asked Questions
🔗 Related Calculators
What is cyclotron frequency?
Cyclotron frequency is the rate at which a charged particle completes one full orbit while moving in a circle inside a uniform magnetic field, given by f_c = zeB/(2*pi*gamma*m). It is the frequency at which a cyclotron's accelerating electric field must alternate to keep pushing the particle forward on every half-turn.
Why is cyclotron frequency independent of kinetic energy at low speed?
At non-relativistic speed, gamma is very close to 1 regardless of how fast the particle is moving, so f_c = zeB/(2*pi*m) stays essentially constant as the particle gains energy and its orbit radius grows. This property is called isochronism, and it is exactly what let the first cyclotrons use a single fixed radio frequency for the entire acceleration cycle.
Why does cyclotron frequency droop at high energy?
As kinetic energy increases, the Lorentz factor gamma = 1 + T/mc2 grows above 1, and since f_c is proportional to 1/gamma, the orbit frequency falls. This relativistic frequency droop breaks isochronism, a fixed-frequency accelerating voltage gradually falls out of sync with the particle's actual orbit period.
Why do real high-energy accelerators use synchrotrons instead of fixed-field cyclotrons?
Once relativistic droop becomes significant, a fixed-frequency, fixed-field cyclotron can no longer stay synchronized with the particle. Synchrotrons solve this by ramping the magnetic field B (and sometimes the RF frequency) in step with the particle's rising momentum, keeping the orbit radius fixed instead of the field fixed, this is the same idea used by synchrocyclotrons and isochronous cyclotrons in different ways.
What units does this calculator use?
Kinetic energy is entered in MeV, magnetic field in tesla, and mass in MeV/c2 (rest energy). The output frequency is reported in megahertz (MHz), period in nanoseconds, and orbital radius in metres.
How is the formula for cyclotron frequency derived?
Equating the magnetic (Lorentz) force zevB to the relativistic centripetal force gamma*m*v^2/r gives r = gamma*m*v/(zeB). Since the orbital period is the circumference divided by speed, T_period = 2*pi*r/v = 2*pi*gamma*m/(zeB), and frequency is the reciprocal, f_c = zeB/(2*pi*gamma*m).
What is the orbital radius formula?
Orbital radius is r = gamma*beta*m*c/(zeB), where beta = v/c and gamma is the Lorentz factor. Radius grows with both kinetic energy and mc2, which is why a cyclotron accelerating particles to higher energy needs a physically larger magnet to contain the growing orbit.
Which particle presets does this calculator include?
Electron (0.51099895 MeV/c2), proton (938.272 MeV/c2), alpha particle (3727.379 MeV/c2, charge 2e), muon (105.658 MeV/c2), and deuteron (1875.613 MeV/c2) are built in, or you can enter any custom rest energy and integer charge.
Why does the electron example show such a dramatic frequency change?
Because the electron's rest energy (0.511 MeV) is tiny compared to a typical kinetic energy of even a few MeV, its Lorentz factor grows huge almost immediately, an electron with only 1 MeV of kinetic energy already has gamma close to 3. Heavier particles like protons need thousands of MeV before the same relativistic droop becomes as dramatic.
What does the frequency vs kinetic energy chart show?
The chart plots cyclotron frequency against kinetic energy from 0 to 5000 MeV at the chosen magnetic field and particle, tracing the classic relativistic droop: flat near zero kinetic energy, then falling as gamma grows. The green dashed marker shows exactly where your current input sits on that curve.
Does a stronger magnetic field change the frequency droop shape?
A stronger field B scales the frequency curve up or down uniformly (f_c is directly proportional to B) but does not change the shape of the droop with kinetic energy, since B cancels out of the ratio between frequencies at different energies. Only the particle's rest energy mc2 changes how quickly gamma, and therefore the droop, sets in.