How it is calculated
An ion of mass \(m\) and charge \(q = ze\) that falls through a potential difference \(U\) gains the kinetic energy \(K = |q|U\). Relativistically its Lorentz factor, speed, and momentum are
The classical values are \(v = \sqrt{2K/m}\) and \(p = mv\). The two agree while \(K \ll mc^2\) (938 MeV for a proton, 511 keV for an electron): the relativistic flight time is longer by about \(\tfrac34 K/mc^2\), 0.04 ppm for Ar⁺ at 2 keV but 0.3 % for an electron at 2 keV.
In a uniform field \(E\) the force \(|q|E\) changes the momentum at a constant rate, both classically and relativistically, so the time to cross a field region is the momentum gained divided by the force. For the two-field source, with the back plate (repeller) at \(V_R\), the extraction grid at \(V_E\), and ground at the second grid,
where \(p_1\) is the momentum at the extraction grid and \(p\), \(v\) the momentum and speed in the drift region. The one-field source is the same with no second gap: \(E_s = U/S\) and \(K = |q|E_s s_0\). The peak spacing for 1 u is the exact difference of two flight times, and every result uses \(|z|\).
Space focus and the Wiley–McLaren condition
An ion that starts farther from the exit grid (larger \(s_0\)) spends longer in the source but gains more energy, so it catches up with an ion that started ahead of it. The catch-up happens at the drift length where the arrival time does not change with \(s_0\) to first order, \(\partial t/\partial s_0 = 0\). The calculator finds it by differentiating the full relativistic \(t\) numerically; classically Wiley and McLaren’s result is
With one field (\(d = 0\), \(k_0 = 1\)) the focus is at \(L_f = 2s_0\), only a centimeter or two from the source: the reason a single-field source cannot space-focus a tube of useful length. With two fields, raising \(V_E\) toward \(V_R\) weakens the source field, raises \(k_0\), and pushes the focus out to the detector. The calculator solves for that extractor voltage by bisection. The arrival spread for a start-position spread \(\Delta s\) is the full range of arrival times over the interval \(s_0 \pm \Delta s/2\), evaluated at 41 points.
An ion born with energy \(\varepsilon_0\) moving away from the detector is stopped and turned around by the source field. It passes its starting point with the same speed as an identical ion that started toward the detector, but later by the turn-around time
No static field arrangement removes it; delayed extraction and reflectrons reduce its effect.
Mass resolution
Because \(t \propto \sqrt{m/z}\) in any electrostatic instrument, \(\Delta m/m = 2\Delta t/t\), so the resolving power is
The calculator adds the timing spread, the arrival spread from \(\Delta s\), and the turn-around time in quadrature, as if each were an independent peak width. That is an estimate: the start-position spread at the focus gives a skewed peak, and the turn-around time is the full separation of the two extreme ions, not a FWHM.
Typical values
Flight times through a 1 m drift region (drift only, singly charged), from the formulas above:
| Particle | m/z | U = 2 kV | U = 20 kV |
|---|---|---|---|
| Electron | 0.000549 | 37.81 ns | 12.27 ns |
| Proton | 1.007 | 1.616 µs | 0.511 µs |
| ⁴⁰Ar⁺ | 39.96 | 10.18 µs | 3.218 µs |
| ¹³²Xe⁺ | 131.9 | 18.49 µs | 5.846 µs |
| C₆₀⁺ | 720.0 | 43.19 µs | 13.66 µs |
| A 10 kDa protein, 1+ | 10 000 | 161.0 µs | 50.90 µs |
Every factor of 10 in voltage shortens the flight time by \(\sqrt{10} \approx 3.16\); every factor of 4 in m/z doubles it. For the electron the relativistic correction is 0.29 % at 2 kV and 2.9 % at 20 kV.
Worked example
The default inputs: ⁴⁰Ar⁺ (m = 39.9618 u, z = 1) accelerated through 2000 V into a 1 m drift tube, with 2 ns of timing spread.
- K = 2000 eV, and \(K/mc^2 = 2000/(39.9618 \times 931.494\times10^6) = 5.37\times10^{-8}\), so γ − 1 is tiny.
- \(v = \sqrt{2K/m}\), with \(2K = 2 \times 2000 \times 1.602\times10^{-19} = 6.408\times10^{-16}\) J and \(m = 39.9618 \times 1.6605\times10^{-27} = 6.636\times10^{-26}\) kg, so v = 98.27 km/s, and t = 1 m / v = 10.176 µs. The relativistic value is longer by 0.04 ppm (0.4 ps).
- An ion of 41 u would arrive 0.126 µs later (t grows as √m: 10.176 µs × (√(41.0/39.96) − 1)). With 2 ns of spread, m/Δm = 10.176 µs / (2 × 2 ns) ≈ 2540.
Switch to Two fields (Wiley–McLaren) to see the source defaults: VR = 2000 V, VE = 1830 V, S = 10 mm, s₀ = 5 mm, d = 10 mm. The source field is 170 V/10 mm = 17 V/mm, so an ion starting at s₀ gains 85 eV before the extraction grid and 1915 eV in all, and k₀ = 1915/85 = 22.5. The formula above gives Lf = 0.991 m, so the detector at 1 m sits 0.9 % beyond the focus; the calculator’s extractor voltage for focus at exactly 1 m is 1830.9 V. The ion arrives at 11.064 µs (0.494 µs in the first gap, 0.172 µs in the second). The 0.5 mm start-position spread adds only 0.24 ns, so with the 2 ns timing spread m/Δm ≈ 2750. An initial energy of 0.04 eV would add a turn-around time of 21 ns and cut m/Δm to about 260: in a linear TOF with a weak source field, thermal motion, not the start position, sets the resolution.
Assumptions and limits
- Ideal grids and uniform fields: no field penetration through the grid meshes, no fringe fields, and no lensing. Real grids transmit 70–95 % of the ions each and slightly deflect them. For electrode shapes, fringe fields, and trajectories, use the Ion Optics Workbench for full field solutions.
- Motion along the axis only. A transverse velocity does not change the flight time in a field-free tube to first order, but it decides whether the ion reaches a detector of finite size.
- No collisions (the mean free path must be much longer than L; see the mean free path calculator), no space charge, and no fragmentation in flight. A metastable ion that fragments in the drift tube keeps its parent’s speed and arrives at the parent’s time in a linear TOF.
- Linear TOF only: reflectrons, orthogonal acceleration, and delayed extraction are not modeled. A reflectron corrects energy spread to first or second order and typically raises the resolving power to 10⁴ and more.
- The source modes assume ions start at rest (except for the turn-around estimate) and that every ion of one m/z experiences the same fields.
References
Primary sources and standards
- W. C. Wiley and I. H. McLaren, “Time-of-flight mass spectrometer with improved resolution,” Rev. Sci. Instrum. 26, 1150–1157 (1955), doi:10.1063/1.1715212 (the two-field source, its space-focus condition, and the turn-around time).
- U. Boesl, “Time-of-flight mass spectrometry: Introduction to the basics,” Mass Spectrom. Rev. 36, 86–109 (2017), doi:10.1002/mas.21520 (one- and two-stage sources, flight-time broadening, and mass resolution).
- J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley (1999), ch. 11–12 (relativistic kinematics; motion in a uniform electric field, where the momentum grows linearly in time).
- P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA recommended values of the fundamental physical constants: 2022,” Rev. Mod. Phys. 97, 025002 (2025), doi:10.1103/RevModPhys.97.025002 (e, u, the electron, proton, deuteron, and alpha-particle masses, and c).
- M. Wang, W. J. Huang, F. G. Kondev, G. Audi, and S. Naimi, “The AME 2020 atomic mass evaluation (II). Tables, graphs and references,” Chin. Phys. C 45, 030003 (2021), doi:10.1088/1674-1137/abddaf (the atomic masses of the preset ions).
Free introductions
- NIST, “Atomic Weights and Isotopic Compositions with Relative Atomic Masses,” nist.gov (isotope masses and abundances, for ions not in the preset list).
- NIST, “CODATA Internationally recommended 2022 values of the Fundamental Physical Constants,” physics.nist.gov/cuu/Constants (the constants used, with their uncertainties).