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) | — |
| Final orbit radius | — |
| Turns to final energy | — |
| Total path length | — |
| Path-weighted mean σ along the orbit | — |
| Beam loss mean free path λ = 1/nσ | — |
| Beam surviving the modeled gas losses over 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 = 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 nearly three 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 = ngaps·q·Vp per turn (every crossing at the voltage peak, as in the path-length calculator), the beam completes ⌊Tfinal/ΔT⌋ whole turns — after k of them it has energy kΔT and the radius of that energy, r(E) = √(2mE)/qB — plus a final partial turn evaluated at its midpoint energy and radius, so the energy used for σ and the energy used for geometry agree at every step. The survival is
The second term is the final partial turn: frac is the fractional turn remaining and Ē its midpoint energy. It vanishes when Tfinal/ΔT is an integer.
Since every term is proportional to n, the optical depth −ln F is linear in pressure (the lost fraction 1 − F is approximately linear only while it is small) 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; outside their range it is held at the end values — an unsourced extrapolation, not data, and the result flags it whenever a lookup leaves the tabulated 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 + H₂⁺ (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. Above that the calculator holds the last value — the measured trend to 50 keV/amu is falling, but whether the held value over- or understates the true loss beyond the table is not established by the source.
| Energy (keV/amu) | σ, H⁺ + H₂ → H + H₂⁺ (cm²) | σ, H₂⁺ + H₂ → total destruction (cm²) |
|---|---|---|
| 1 | 4.26 × 10⁻¹⁶ | — |
| 1.5 | — | 8.12 × 10⁻¹⁶ |
| 2 | 6.88 × 10⁻¹⁶ | 8.18 × 10⁻¹⁶ |
| 4 | 9.33 × 10⁻¹⁶ | 8.30 × 10⁻¹⁶ |
| 7 | 9.57 × 10⁻¹⁶ | 8.15 × 10⁻¹⁶ |
| 10 | 8.66 × 10⁻¹⁶ | 8.18 × 10⁻¹⁶ |
| 15 | — | 8.46 × 10⁻¹⁶ |
| 20 | 5.79 × 10⁻¹⁶ | 7.75 × 10⁻¹⁶ |
| 40 | 2.50 × 10⁻¹⁶ | 4.69 × 10⁻¹⁶ |
| 50 | — | 3.81 × 10⁻¹⁶ |
| 70 | 7.78 × 10⁻¹⁷ | — |
| 100 | 2.91 × 10⁻¹⁷ | — |
| 200 | 1.76 × 10⁻¹⁸ | — |
| 400 | 6.09 × 10⁻²⁰ | — |
| 700 | 3.64 × 10⁻²¹ | — |
| 1000 | 4.14 × 10⁻²² | — |
Simple mode is exact when handed the right path-weighted σ for the trajectory, and conservative only when handed a true upper bound over it — which "the low-energy end" is not automatically, since the proton cross-section rises from 1 to 7 keV/amu. For the reference 2–150 keV proton spiral the two modes agree closely (its path-weighted σ lands near 10⁻¹⁶ cm²), but that agreement is the trajectory's and does not transfer: a 20 keV, 1 kV proton machine's path-weighted σ is 7.5 × 10⁻¹⁶ cm², and the default 10⁻¹⁶ would permit about seven times too much pressure there. For H₂⁺ the default understates the loss several-fold at any tabletop energy — the destruction cross-section stays within a factor of about two of its peak across the table's 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.
- Orbit mode's stopping convention: whole turns at their own energy and radius, then one partial turn evaluated at that segment's midpoint energy and radius — an approximation, not a gap-by-gap orbit integration.
- 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. That can be optimistic: dee conductance limits and the source's local gas load usually make the beam-path pressure higher than the gauge reads. The vacuum page quantifies the dee-interior excess.
- The cross-sections are for hydrogen gas, and the calculator enforces it: orbit mode locks the gas selector to H₂. Air cross-sections differ and no air loss model is offered.
- The result is survival against the modeled gas-loss processes only — capture for protons, total destruction for H₂⁺. Source efficiency, phase loss, scattering acceptance, centering, and extraction are not modeled.
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% — the survival arithmetic behind the 10⁻⁵–10⁻⁶ torr operating pressures in the census's record (which pressure a given machine needed also turned on RF holdoff, source load, and how much loss it could accept).
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 — 37.5 turns, 14.9 m — survives 71.5% at 1 × 10⁻⁵ torr, where simple mode with 10⁻¹⁶ cm² would have said 95.1%. Both values are recomputed against an independent implementation whenever this page's model changes; the last such revision (energy-consistent stopping) is dated in the changelog.
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.
Educational reference, not an operating procedure: results reflect the stated model and assumptions. Verify anything safety-critical against primary sources, and read the safety fundamentals before applying numbers to real hardware. Last reviewed: · Report a correction.