Cyclotron / Mass Spec

Compare ion paths and detector hits, or explore energy gain in an ideal RF cyclotron.

About this tool

Use magnetic curvature to compare mass-to-charge ratios in a mass spectrometer, or follow successive energy gains in an ideal radio-frequency cyclotron. These are two separate modes with different particle paths.

Inputs and conventions

B is the signed magnetic field component perpendicular to the picture, in tesla. Positive B points out of the plane. Use 0.01 ≤ |B| ≤ 2.5 T. Particles enter at (0, 0), moving along +y at 20–500 km/s. A positive charge bends right for positive B. Changing the sign of B or the charge mirrors the bending direction.

Each particle has an editable mass from 1 to 250 u and an integer charge number z from −5 to +5, excluding zero. Its mass in kilograms is m = mass in u × u, and its charge is q = ze. This model requires nonzero charge and a nonzero magnetic field because it describes magnetically curved trajectories.

The proton, deuteron and alpha presets use CODATA 2022 ion masses. The alpha particle has mass 4.001506179129 u and charge +2; it is a helium nucleus, not a neutral helium atom. Particle presets keep the chosen field and entry speed. Cyclotron presets also keep the voltage and number of half-circles.

Use and time

Select Render to prepare the complete paths. Physical time t is in microseconds. The time field and slider move the markers together without changing the calculated trajectories. Start shows the full physical flight over five screen seconds; Pause, Resume and Repeat control this slow-motion view. The displayed time is physical time, not elapsed screen time.

Changing parameters clears the result and starts the next rendering at t = 0. An invalid time hides current markers and values while retaining the valid paths. Clear result keeps the parameters. Leaving or hiding the tool pauses playback; returning within the app keeps the paused result and selected time. Reloading restores valid settings and waits for Render.

Magnetic motion

The classical angular frequency is ω = |qB|/m, the frequency is f = ω/(2π), the period is T = 1/f, and the radius is r = v/ω. Kinetic energy is K = mv²/2. The graph uses the same meter scale in both directions; the center and radius line belong to the current circular arc.

Mass spectrometer

Particles A and B leave an ideal velocity selector at the same entered speed v. The selector uses Bs = 0.1 T and requires electric-field magnitude |Es| = vBs. The force balance selects speed independently of mass and charge magnitude. The analyzer then uses the separate signed field B from the input; selector values are listed under More values.

With s = sign(qB), the analyzer path is x = sr(1 − cos θ), y = r sin θ, where θ = ωt from 0 to π. The detector is y = 0, and the hit is x = 2sr after flight time T/2. The detector separation is |2sA rA − 2sB rB|. The graph marks the entry, both complete half-circles and both hit positions.

The examples compare proton/deuteron, masses 2 u and 4 u at charge +1, equal m/|z| using 2 u/+1 and 4 u/+2, and opposite charges using 2 u/+1 and 2 u/−1. Equal mass-to-charge ratios with matching signs produce the same trajectory; reversing one charge mirrors its trajectory.

The shared time interval ends at the later detector hit. An earlier marker remains at the detector. Listed velocity at and after a hit is the incoming velocity, not motion along or beyond the detector.

RF cyclotron

The cyclotron follows particle A through N = 1–20 half-circles. The entered speed is its speed on entering the first dee. Each of the N − 1 internal gap crossings adds ΔK = |q|U, with gap voltage U from 0 to 2000 V. The model treats these crossings as instantaneous and the gap as having zero width. Within each dee, only the magnetic field changes the direction of motion.

For half-circle j = 0…N − 1, Kj = K0 + j|q|U, vj = √(2Kj/m), and rj = vj/ω. Its duration is T/2 even as energy and radius increase. The radio frequency is matched to this ion at fRF = f, and the voltage polarity reverses every half-circle. Different ion species are not accelerated together by a single common resonance in this mode.

Let σj = (−1)ʲ and let half-circle j start at (xj, 0). Its center is (xj + sσj rj, 0), and its path is x = xj + sσj rj(1 − cos φ), y = σj rj sin φ, with φ from 0 to π. The next arc starts at xj+1 = xj + 2sσj rj. The lightly shaded regions above and below y = 0 identify the two dee half-planes; the circle center can change between arcs.

Exactly at an internal gap crossing, the time probe shows the state just after the energy impulse. At the final endpoint it shows the incoming state of the last half-circle. U = 0 keeps the radius constant, and N = 1 gives no additional energy impulse.

Assumptions and sources

The model is classical and nonrelativistic, with uniform fields, vacuum, no fringe fields and no interactions between particles. It is an idealized trajectory comparison, not a model of detector response or accelerator losses. The constants are u = 1.66053906892 × 10⁻²⁷ kg and the exact elementary charge e = 1.602176634 × 10⁻¹⁹ C.