Read the coordinates before the colors
Horizontal axis: q describes the RF term. Vertical axis: a describes the DC term. For a positive ion in this convention, a = 4zeU/(mr₀²Ω²) and q = 2zeV/(mr₀²Ω²); U and V are pair-to-pair, and V is zero-to-peak. The slope is k = a/q = 2U/V, not U/V. See the symbol and unit definitions.
Start with the diagram, then try the guided experiments. The equations below explain the boundary labels when you need them.
Stability interiors and marginal boundaries
The canonical Mathieu equation is shown below. Shaded interiors indicate ideal infinite-time stability. Characteristic curves mark marginal boundaries, where bounded motion is not guaranteed for every initial condition.
In the mapping equation, Lₙ(q) and Uₙ(q) mean the lower and upper characteristic boundaries of the nth X band. They are boundary functions, not the electrode voltages U and V. The plotted q is nonnegative; the RF-phase symmetry is already included when Y is mapped into this frame.
The first-quadrant expression requires a positive upper bound. The a = 0 axis is a plotting cut through the full symmetric region. At a = q = 0 there is no RF confinement: an ion with transverse velocity drifts. Curves are numerically evaluated and sampled; root classification is independent of drawing resolution. See NIST DLMF stability theory and the canonical Mathieu equation.
Interactive diagram
Predict what will happen, enter a slope, then compare the two numerical boundaries. The number field is useful near the apex; “Fit passband” magnifies a valid interval without changing the calculation.
Guided experiments: predict, change, explain
The numbers below are rounded checks of this ideal model, not measured instrument specifications. The solver uses unrounded values; the endpoints themselves are marginal and excluded.
1. Watch a passband shrink, then disappear
Begin at k = 0.33. Record entry and exit. Try 0.3356, then use “Fit passband.” Finally enter 0.336. Before each change, predict whether the two crossings will move closer together, exchange order, or vanish from the screen.
| Slope k | Entry q | Exit q | Interpretation |
|---|---|---|---|
| 0.33 | 0.692840636 | 0.708700703 | A finite ideal interval |
| 0.3356 | 0.705793383 | 0.706037489 | A narrow but nonzero interval |
| 0.336 | No simultaneous first-region interval | The separate boundary crossings do not enclose a valid overlap | |
Why is an empty-looking sliver at 0.3356 different from the result at 0.336?
At 0.3356 the mapped-Y entry is below the X exit, so some q values satisfy both conditions; the interval may be thinner than a display pixel at this zoom. At 0.336 there is no common first-region interval. Drawing resolution does not decide mathematical stability. The readout is calculated from roots independently of the plotted samples.
2. Test the apex without rounding it
Click “Use apex slope.” It uses the calculated slope near 0.335687411416740. Is touching the overlap at one point the same as having a transmitting band?
Reveal the apex explanation
No. The entry and exit coincide at a marginal boundary point, leaving zero interval width. Do not describe it as an infinitely resolving, guaranteed-transmission setting. A real instrument also has finite aperture, injection distributions, and finite transit time; the diagram alone does not predict its measured line shape or signal.
3. Separate display changes from field changes
Return to k = 0.33 and reduce the visible q extent to 0.30. Then use “Fit passband.” Toggle the X and mapped-Y layers one at a time.
Did hiding a crossing or a layer change which ions are stable?
No. The roots remain near 0.693 and 0.709 when they are off-screen. Zoom and layer toggles change the view only. Both X and mapped Y must be tested against the same positive scan line; reflecting that line as well would change the physical interpretation rather than reveal an additional requirement.
4. Set the DC term to zero
Enter k = 0. The first simultaneous RF-only interval is 0 < q < 0.908046334. At fixed RF amplitude and frequency, q increases as m/|z| decreases.
Which side is the low-mass cutoff? Does the origin describe a confined guide?
The high-q boundary sets the low-mass edge of the first RF-only interval. The ideal first-region picture does not provide a finite upper mass cutoff along a = 0, but real transport remains limited by injection, field strength, aperture, and other conditions. At the origin the transverse equation becomes free drift; nonzero transverse velocity is not confined.
What the result does and does not tell you
- The first quadrupole mass-filter operating region is the overlap of X and Y first-region stability after mapping the Y-axis equation into the x-frame.
- Higher Mathieu stable regions are plotted for completeness and intuition, but conventional quadrupole mass filters typically operate using the first-region overlap.
- The passband has two boundaries: mapped-Y entry at
−a₀(q) = kqand X exit atb₁(q) = kq. The interval must have positive width. Its near-apex limit isk ≈ 0.335687411416740. - Ideal stability does not promise transmission. Aperture, injection conditions, RF phase and finite transit time matter; an unstable trajectory can also survive a short device.
- The first-region root calculations are independently checked against characteristic-value calculations; the graphic remains a sampled illustration. SciPy characteristic values provide independent regression checks.
Translate the plot back into instrument language
At fixed voltages, larger m/|z| moves toward lower q. Therefore the q exit corresponds to the lower-mass edge and the q entry to the higher-mass edge. The worked mass-window example shows the conversion. A q-interval width is not a measured FWHM, and the diagram has no detector or ion-abundance model.
Model boundary: this is ideal, collisionless, infinite-time transverse stability in a sinusoidal quadrupole field. It excludes finite rod ends, mechanical aperture, injection phase-space distributions, space charge, and collisions. Higher one-axis bands shown in the overview are not automatically higher simultaneous transmission bands; this explorer's passband readout calculates the first X/Y overlap only.