Vacuum & Beam Survival Calculator
A cyclotron beam dies by collision: every residual gas molecule along the spiral is a chance to charge-exchange or scatter out of the machine. This calculator converts a chamber pressure into a mean free path and an estimated surviving beam fraction over your total path length (from the path length calculator), and tells you what pressure keeps losses under 10%.
Results
| Pressure (all units) | — |
| Gas number density n | — |
| Gas–gas mean free path (kinetic theory) | — |
| Beam loss mean free path λ = 1/nσ | — |
| Estimated beam surviving L | — |
| Pressure for <10% loss over L | — |
The math
From the ideal gas law, the number density of residual gas molecules is
A fast beam particle traversing (effectively stationary) gas with a loss cross-section σ has a mean free path λ = 1/nσ, and — in the simple single-collision-loss model used here, where every such collision removes the particle — the fraction surviving a path L is
Requiring F ≥ 0.9 gives the pressure bound p ≤ −ln(0.9)·kBT/(σL). For reference the calculator also shows the ordinary kinetic-theory mean free path of the gas itself, λgas = kBT/(√2 πd2p), using hard-sphere diameters d = 2.71 Å for H₂ and 3.66 Å for air — note the √2 applies only to gas–gas collisions, not to the fast beam.
Choosing the cross-section
For protons below ~100 keV the dominant loss is electron capture (charge exchange); the neutral atom no longer bends in the field and flies into the wall. The cross-section falls steeply with energy — rough values for H⁺ in H₂ gas (Barnett et al. 1990; Allison 1958):
| Proton energy | σ (charge exchange, H₂) |
|---|---|
| 10 keV | ~8 × 10⁻¹⁶ cm² |
| 30 keV | ~5 × 10⁻¹⁶ cm² |
| 100 keV | ~5 × 10⁻¹⁷ cm² |
| 300 keV | ~1 × 10⁻¹⁸ cm² |
Because the beam spends many turns at low energy, where σ is largest, using a cross-section from the low-energy end of your machine's range is the conservative choice — this site recommends it. H₂⁺ beams additionally break up by collisional dissociation at a similar 10⁻¹⁶ cm² scale.
Assumptions and limits
- Single constant σ over the whole spiral; in reality σ varies orders of magnitude with energy. The exponential model is stated as such — treat results as an estimate, not a prediction.
- Every loss collision removes the particle; small-angle scattering that merely dilutes the beam is not counted separately.
- Uniform pressure and gas composition throughout the chamber.
Worked check
At 1 × 10⁻⁵ torr of H₂ at 20 °C, n = 3.29 × 10¹⁷ m⁻³. With σ = 10⁻¹⁶ cm² (10⁻²⁰ m²), λ = 304 m; over L = 7.6 m survival is 97.5%. With a low-energy σ = 10⁻¹⁵ cm² and L = 15 m, λ = 30.4 m and survival drops to 61% — which is why historic classical cyclotrons ran in the 10⁻⁵–10⁻⁶ torr range.
Sources
- C. F. Barnett et al., Atomic Data for Fusion, Vol. 1: Collisions of H, H₂, He and Li Atoms and Ions with Atoms and Molecules, ORNL-6086, 1990 — charge-changing cross sections.
- S. K. Allison, "Experimental Results on Charge-Changing Collisions of Hydrogen and Helium Atoms and Ions", Rev. Mod. Phys. 30, 1137 (1958).
- J. F. O'Hanlon, A User's Guide to Vacuum Technology, 3rd ed., Wiley, 2003 — kinetic theory, molecular diameters.