Bainbridge mass spectrometer

Follow the sequence from physical construction to a defensible mass-to-charge measurement. Each experiment includes an objective, assumptions, controls, procedure, measurement table and interpretation questions.

Objective

Identify every stage and distinguish the instrument geometry from the model assumptions.

Objective

Measure the relationship between accelerating voltage and ion kinetic energy.

Objective

Verify circular motion, radius and kinetic-energy conservation.

Objective

Use the cross product and charge sign to predict beam deflection.

Objective

Measure the transmitted velocity band and off-design rejection.

Objective

Recover m/|q| from detector position and quantify the change caused by nonideal fields.

Objective

Separate isotopes and compare measured abundances with source composition.

Objective

Infer an unknown m/|q| using only modeled instrument measurements.

Objective

Compare the two different radius-to-mass relationships without interchanging them.

Objective

Investigate the tradeoff between slit acceptance and peak width.

Objective

Separate detector broadening, random sampling and numerical repeatability.

Objective

Use physical diagnostics to distinguish meaningful failure conditions.

Ion Lab is an original educational Bainbridge instrument, not a replica of a named commercial mass spectrometer. Its purpose is to connect inspectable hardware, local Lorentz forces and measured detector events. All geometry dimensions are design assumptions unless a data source is explicitly named.

Coordinates and dimensions

One scene unit equals one metre. The right-handed coordinates are +x along the incident beam, +y toward the positive selector electric field, and +z through the pole gap. For a positive ion moving +x in +Bz, v × B points −y. The ion plane is z = 0. The detection surface is x = 0 on the returning side of the orbit.

Two source preparations, two calibrations

Full source: a parameterized ion-production layer at x = −0.520 m samples isotope, transverse position, direction and initial kinetic energy. The gas inlet and filament gate production. Source rate is the modeled post-ionization ion rate, not a calculated electron-impact cross section. Emission times are regularly spaced at 1/rate; isotope identity is independently sampled categorically, y/z coordinates and angular slopes are Gaussian, and charge state is fixed for a run. Temporal shot-noise currents are not modeled. Twelve percent of the source potential drops over 35 mm and 88% over the next 120 mm. Both fields are integrated, so acceleration depends on signed qE.

Ideal injected beam: ions start at x = −0.351 m with prescribed speed and Gaussian transverse spreads. Upstream source hardware is present for inspection but bypassed explicitly. This mode does not claim to simulate ionization. The signed sandbox is also a prescribed synthetic beam.

Constant-energy mode: the source remains active and both selector field contributions are zero. Electrodes and apertures still intercept particles. With negligible initial energy, the correct calibration is m/|q| = Bₐ²r²/(2U), not the equal-speed relationship.

Derivations and assumptions

Electrostatic work gives ΔK = q(φin − φout). With negligible initial energy and accelerating gain |q|U, v = √(2|q|U/m). Crossed fields select the undeflected direction when E + v × Bₛ = 0; for the reference orthogonal arrangement v₀ = E/Bₛ. Equating the perpendicular magnetic force |q|v⊥Bₐ to mv⊥²/r gives r = mv⊥/(|q|Bₐ). A nominal half-circle has detector displacement |y| = 2r. Therefore, at selected velocity m/|q| = BₐBₛr/E; with fixed acceleration energy, substitution of v² yields m/|q| = Bₐ²r²/(2U). Finite entrance position, divergence, broad acceptance and fringe fields invalidate exact half-circle inference.

Ideal and finite-geometry field models

Ideal: hard-edge uniform field regions are suitable for textbook benchmarks. The source uses an axial piecewise-linear potential. The selector has uniform Ey = Vs/d and Bz = Bₛ within its working gap. The analyzer has uniform Bz = Bₐ. These clipped field regions are idealizations; the selector hard edges are not a global electrostatic boundary-value solution.

Finite geometry: the selector electric field comes from the exact Coulomb surface integral of two finite uniformly charged rectangular sheets, scaled so their centre-to-centre surface potential difference is the selected voltage. Magnetic fields are exact free-space integrals of paired finite equivalent pole sheets, each normalized to its own magnet-centre field. Their contributions superpose, including transverse and fringe components. This is a physically defined finite-source surrogate: the electric sheets are not solved equipotential conductors and the magnetic sheets are not a nonlinear iron/yoke solution. Source acceleration remains the same reduced 1D model. No arbitrary visual falloff is used.

The integrator, arrows, field lines, magnitude map, numerical probe and telemetry all query the same Fields.at(x,y,z) function. Field lines are numerical integral curves of the static vector field; they are not material wires or propagating electromagnetic waves. Arrow lengths are normalized for visibility and the section map is sampled at z = 0.

Integrator, boundaries and deterministic timing

The synchronized second-order drift–Boris-rotation–drift method maintains x and v at the same physical timestamp. It applies a half electric impulse, an orthogonal Cayley magnetic rotation and a second half electric impulse at the drift midpoint. This avoids mixing leapfrog half-step velocities with full-step positions. Uniform magnetic rotation preserves kinetic energy up to floating-point roundoff.

The acquisition clock advances in exact 50 ns quanta, independent of rendering. Each particle substeps by at most 2 ns by default, no more than 0.005 radians of local cyclotron rotation, and no more than 0.2 mm of travel. Longitudinal ideal-field edges subdivide the step. Every candidate path segment is tested against slit, electrode, roof and wall bounds, and the earliest intersection is used. Curved motion between substeps is approximated by the chord; the executable convergence tests document its error. Detected-state interpolation is a small additional approximation and is not the 100-period energy benchmark.

Random streams are seeded independently for each ion and detector response. Display visibility never consumes a physical random number. Acquisition duration controls emission; already emitted ions are allowed to finish. An 80 µs individual observation horizon is explicitly counted as termination, never silently discarded. Reproducibility is tested in the development Node and browser environment; cross-engine transcendental functions can differ in their last floating-point bits.

What the detector measures

A count is created only when an integrated trajectory crosses the active return surface within its y/z acceptance. A Bernoulli trial implements detection efficiency; Gaussian position noise uses the stated σ; the measured y coordinate is assigned to an electronic segment. Out-of-range readouts are counted as undetected, not clamped. A zero-σ, 16,384-segment validation setting explicitly enables continuous position readout; all other settings use quantization.

All events have unit weight. Raw counts are Monte Carlo ion events, not a simulated continuous analog signal. Changing source rate changes the number of emitted ions; changing visible particle fraction changes only what is drawn. There are no predefined isotope peaks, species-driven mass axes, additive decorative noise or forced abundance ratios.

The ideal mass axis uses only detector position and operating settings. In finite geometry, it remains unavailable until synthetic known standards produce a monotonic measured calibration. Linear interpolation is allowed only within the measured range; there is no extrapolation. Calibration uses a zero-spread beam under the same field/geometry settings, so a broad unknown beam can have bias and broadening not removed by the calibration. The cross-validation test reports held-out reference error; this is not experimental traceability.

Statistics, peak widths and uncertainty

Transmission is the fraction of emitted ions crossing the selector exit acceptance plane; it is a cumulative subset, not an extra terminal category. Overall detection efficiency is detected/emitted. The balance is emitted − active − detected − rejected − lost − undetected − terminated, and must be zero.

Peak detection uses the measured histogram, with a [0.2, 0.6, 0.2] three-bin smoothing kernel for the local maximum and half-height crossings only. The chart itself displays unsmoothed counts. Reported FWHM and resolving power are histogram estimates; peaks below two-bin width are flagged as bin-limited. Global mean and sample standard deviation are not individual isotope peak statistics. Sparse or overlapping peaks may not support a useful FWHM. Source and detector distributions use Gaussian standard deviations; injected energy is truncated to remain positive. In physical-source modes the energy-spread percentage applies to the initial source energy, not an imposed spread in accelerating voltage. Statistical standard errors are sample σ/√N; they exclude field-setting and calibration-systematic uncertainty. For mixtures they describe the global distribution, not individual peaks. A broad beam can yield a very small standard error around a systematically biased mean.

Visualization and inspection

Component reference dimensions describe the working aperture, active face or named subassembly, not necessarily its complete mechanical envelope. The instrument uses original procedural solid geometry rendered by the included LabGL WebGL 2 micro-renderer. Metal, copper and ceramic appearances are illustrative material shading, not measured optical constants. Exterior, interior, independent X-ray transparency, interactive x/y/z clipping, explosion, component focus, orthographic/perspective and particle-follow views are provided. Explosion pauses the experiment and hides trajectories to avoid representing moved mechanics as assembled physics. Section-cut edges are highlighted; there is no claim to model physical X-ray imaging.

Ions are rendered as enlarged markers, vector lengths are normalized, trajectories use bounded history buffers and physical time is greatly slowed. Playback is a target maximum; rendering load may slow wall-clock presentation without changing the numerical step. The always-visible selected ion is exempt from the visual-density filter. None of these visualization aids changes the particle properties, SI geometry, detector data or collision radii. A dashed analytical reference is separately labeled and is never used by the integrator.

Sources and data provenance

NIST neon masses and isotopic compositions:19.9924401762 u (0.9048), 20.993846685 u (0.0027), 21.991385114 u (0.0925).NIST database provenanceidentifies AME2012 masses, 2009 isotopic compositions, and the January 2015 dataset update. Retrieved 2026-10-03 by the instrument’s author. Equal-fraction mixtures and all synthetic masses are explicitly educational choices.

Ion masses subtract z × 0.0005485799090441 u from neutral masses. Ionization-energy/c² corrections are omitted and are negligible for this laboratory’s stated benchmark targets. SI conversion uses u = 1.66053906892 × 10⁻²⁷ kg, e = 1.602176634 × 10⁻¹⁹ C exactly, and c = 299792458 m/s exactly.CODATA 2022 constants (NIST)

Conceptual references:OpenStax, Bainbridge mass spectrometry·OpenStax, charged-particle magnetic motion·PlasmaPy’s Boris integrator documentation·The equations are implemented in original code; no textbook artwork, prose excerpts or external software is bundled.

Export and software organization

The physics kernel contains the field definitions, sampling, integration, collisions, detector response, calibration and measurement statistics, and has no rendering dependency. The renderer holds independent geometry and WebGL drawing. The validation suite runs in this browser from the Validation dialog. Full configuration JSON stores seed, physical model, detector settings, and calibration; it replays from t = 0 rather than restoring an arbitrary live particle state. Manually timed emission actions are not an exported schedule; exact replay is defined for a standard full acquisition from t = 0.

A graphics failure is reported as a fault; no hidden scientific approximation is substituted for the missing renderer, and the measurements already recorded stay available.

Three working regions explain the instrument. Reveal them selectively rather than filling the chamber with luminous lines. Every displayed curve and arrow samples the very same Fields.at(x, y, z) object as the charged-particle integrator.

Advanced layer / entrance and exit fringes

Use Fringe fields only with the finite-sheet model. Curves are integrated through the finite charge-sheet and magnetic pole-sheet representation. They expose nonuniformity and make retuning, acceptance and calibration worth investigating. The ideal model draws no invented fringes. Choosing a region never changes the experiment; loading the explicitly labelled finite-field preset does.

What must not be implied by the picture

B lines are continuous in the physical field. The short gap segments terminate at a display boundary, not at a magnetic monopole. This model does not solve the nonlinear steel return yoke, so it does not draw fabricated closed return-flux paths through that steel. It also does not invent electron-impact source fields, slit focusing fields or detector collection fields that are absent from the kernel.

Suggested teaching sequence

Inspect materials → load Full source and inspect E → return to equal-velocity injection and inspect the selector → emit one ion and compare force arrows → use Beam plane for the analyzer → collect the measured spectrum → compare finite-field fringes and calibrated measurements.

References:OpenStax: velocity selector and mass spectrometer·OpenStax: charged-particle motion·OpenStax: magnetic field lines

The instrument is an educational engineering design, not a scanned commercial product. These material identities follow plausible laboratory construction. Their shader parameters are appearance estimates, not a measurement of a particular real instrument.

How the surfaces are rendered

A metal–dielectric microfacet BRDF, Fresnel reflection, anisotropic machining grain, differentiated roughness and enamel clear coat respond to a neutral analytical studio environment. Derivative filtering limits fine-grain aliasing. A depth shadow map supplies contact and cast shadows. The workspace is dark, but the instrument is illuminated so its steel, copper and ceramic remain distinguishable.

Inspect without teaching overlays

Select Materials only above the viewport. Exterior, Interior, Section, X-ray, Exploded, component isolation and close-up focus remain available. X-ray opacity and clipped section highlights are explicit inspection aids, not the natural appearance of the real materials.

Construction references:Vacuum-hardware supplier data: stainless CF flanges and OFHC copper seals·Vacuum-component supplier data: alumina/metal vacuum feedthroughs·Remaining construction and finish choices are declared design assumptions; these sources do not identify this educational instrument.

The exports separate measured detector readout from internally known simulator quantities. All recorded events have unit statistical weight.

“Save experiment” exports the complete physics configuration and seed for replay from t = 0. It is not a snapshot of live integrator states. The files are created in your browser and are not uploaded.

Each sweep point starts an independent acquisition with the same seed and a new static setting. No magnet-ramping or induced-field claim is made. The current workspace acquisition is preserved.

Changing the source preparation, charge, active field settings, slit geometry or numerical step invalidates this calibration. Broad beams can retain preparation-dependent systematic bias.

All count data originate in integrated ion trajectories. Transmitted is a cumulative subset; it is not added to the terminal population balance.

Measured beam and detector statistics

Standard errors describe repeated-sample precision only, not field or calibration uncertainty. For a mixture these are global statistics, not uncertainties for individual isotope peaks.

Measured mass-spectrum peaks

No adequately sampled calibrated peaks are available.

The displayed histogram is unsmoothed. A [0.2,0.6,0.2] three-bin smoothing kernel is used only to estimate maxima and half-height crossings. Sparse or overlapping peaks can make FWHM misleading. Global mixture standard deviation is not an individual peak width.

Physical loss / termination reasons