Vacuum & Beam Survival Calculator

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)·k_BT/(σL). For reference the calculator also shows the ordinary kinetic-theory mean free path of the gas itself, λ_gas = k_BT/(√2 πd²p), using hard-sphere diameters d = 0.274 nm for H₂ and 0.366 nm for air (O'Hanlon App. B.2) — note the √2 applies only to gas–gas collisions, not to the fast beam.

Along the orbit

The cross-section is not constant: for protons in hydrogen it peaks near 10⁻¹⁵ cm² at 7 keV and falls two orders of magnitude by 200 keV, so a beam loses most of its particles in its early turns. Orbit mode replaces the single σ with the sum over turns. With ΔT = n_gaps·q·V_p per turn (every crossing at the voltage peak, as in the path-length calculator), the beam reaches its final energy in N = T_final/ΔT turns; after k turns it has energy kΔT and radius r_k = r_final·√(k/N), with r_final = √(2mT_final)/qB. The survival is

Since every term is proportional to n, the loss is still linear in pressure and the curve above keeps its shape; what changes is the effective σ, reported as the path-weighted mean. σ(E) is interpolated log-log in the tables below and held at the end values outside their range. For H₂⁺ the tables are tabulated per nucleon, so a 150 keV H₂⁺ ion is looked up at 75 keV/amu.

The cross-sections

Recommended values from the Oak Ridge compilation (Barnett et al. 1990, ORNL-6086 Vol. 1; OSTI full text, retrieved 2026-08-23). Protons lose by electron capture, H⁺ + H₂ → H (p. A-28; stated accuracy 10% above 100 eV). H₂⁺ is destroyed by capture and by collisional dissociation together; the total-destruction cross-section (p. G-46; accuracy 20%) is tabulated only to 50 keV/amu, and above that the calculator holds the last value, which overstates the loss.

In simple mode, because the beam spends many turns at low energy where σ is largest, a value from the low-energy end of the machine's range is the conservative choice. For a proton machine the two modes agree closely — across a 2–150 keV spiral the path-weighted σ lands near 10⁻¹⁶ cm² — but for H₂⁺ the simple default understates the loss several-fold, because the destruction cross-section never falls within a tabletop machine's energy range.

Assumptions and limits

• Simple mode: one constant σ over the whole spiral. Orbit mode: σ(E) per turn from the tables, every gap crossing at peak voltage, orbit radius from the non-relativistic energy formula.
• 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 — in particular, the pressure inside the dee is taken equal to the chamber pressure. The vacuum page explains when that fails.
• The cross-sections are for hydrogen gas. Air is the residual gas of a leaking or outgassing chamber and its cross-sections differ; the tables are not applied to it.

Worked checks

Simple mode: 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.

Orbit mode, the site's reference point: protons, B = 0.6 T, 150 keV, 1 kV peak on the dee, two crossings — 75 turns to a 9.33 cm radius, 29.6 m of path. At 1 × 10⁻⁵ torr survival is 89.9%, against 90.7% from simple mode with σ = 10⁻¹⁶ cm². H₂⁺ to 75 keV at the same radius — 38 turns, 15.1 m — survives 71.3% at 1 × 10⁻⁵ torr, where simple mode with 10⁻¹⁶ cm² would have said 95.1%. Both were computed independently in Python from the tables above before the page shipped.

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/V1, 1990 — H⁺ + H₂ capture p. A-28, H₂⁺ + H₂ total destruction p. G-46. OSTI, retrieved 2026-08-23.
• J. F. O'Hanlon, A User's Guide to Vacuum Technology, 3rd ed., Wiley, 2003 — kinetic theory; molecular diameters from Appendix B.2 (H₂ 0.274 nm). Library entry.

The narrative behind this calculator — gas load, effective pumping speed, the dee interior, and a worked budget — is the vacuum page; the sourced rules are the design guide’s vacuum domain. What gas scattering does to the beam itself is covered in beam quality.