Velocity Selector and Mass Analyser

What is modelled

Singly charged ions leave the source along x. Between two plates a uniform electric field E and magnetic field B_s are perpendicular to the beam and to each other, so the sideways force q(E − v B_s) vanishes only at v = E/B_s. Faster or slower ions drift on a cycloid and miss the exit slit. Behind the slit a uniform field B_a turns the ions through a half circle of radius r = m v / (q B_a), so heavier isotopes land farther from the slit.

Equations

• Lorentz force F = q (E + v × B); filter speed v = E / Bs.
• Cycloid in crossed fields: drift speed E/B, offset y(t) = −(v0 − E/B)(1 − cos ωt)/ω, ω = qB/m.
• Sector radius r = m v / (q Ba); landing distance 2r; separation of two masses 2 Δm v / (q Ba).

Checks run in the page

• Half-circle diameter from the Boris integrator against 2 m v / (q B).
• Numerical offset at the slit plane against the analytic cycloid.
• Mass dispersion between 20, 22 and 24 u against theory.
• Kinetic energy conserved by the magnetic rotation.

Honest simplifications

Ideal uniform fields with sharp edges, no fringe fields, no space charge or collisions, a point source and a purely geometric slit. The numbers follow the 3D rig’s scale and illustrate the principle, not a specific instrument.

• Press Selector off: ions of all speeds enter the slit and the pattern broadens.
• Press Wide speed spread and watch the slit trim the beam.
• Raise the analyser field and see the species crowd closer together.