---
title: "How four straight rods select ion mass"
description: "Follow ions through alternating focus to see how rod voltages, finite flight and known-ion calibration turn collector hits into a mass spectrum."
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  zh-CN: "https://sandboxphysics.com/zh/stories/instruments/quadrupole-mass-filter/"
last_modified: "2026-11-11"
---

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Inside the instrument

# How four straight rods select ion mass

A quadrupole selects mass-to-charge ratio through transverse motion. Rapidly reversing rod voltages let a range of ions reach the collector before hitting metal. Scanning the voltages moves that range through a mixed beam.

Sandbox Physics · 2026-11-11

## A narrow path to the collector

An ion enters the space between four parallel metal rods. A collector waits at the far end. The ion can contribute to the signal only if it reaches that collector before striking metal. A quadrupole mass filter uses this journey to select part of a mixed beam.

The quantity it selects is mass-to-charge ratio: mass divided by electric charge. The rods remain straight, but the ions move sideways as they travel along them. Sweep the rod voltages and different mass-to-charge ratios reach the collector in turn. Their collected signal forms a spectrum.

Sources: [2]

## A static field loses one direction

Give one opposite pair of rods a positive voltage and the other pair a negative voltage. A positive ion is pushed toward the centre along one transverse direction and pulled away along the other. Static focusing in one direction comes with defocusing in the other.

Wolfgang Paul began with this picture in his 1989 Nobel lecture. Reverse the voltages periodically and the two directions exchange roles. The ion experiences alternating focusing and defocusing. The field also grows stronger farther from the centre, so its motion over a complete cycle can repeatedly bring it back. The ion has changed position and velocity before the next half-cycle. The next force acts at that new position, so the complete motion can accumulate a restoring effect.

The timing and strength of the drive matter. Under suitable conditions the ion can continue down the axis. Under others, an outward excursion grows until the particle reaches a rod.

Sources: [1]

## Mass changes the response

A heavier ion with the same charge accelerates less in the same electric field. During one voltage cycle it changes transverse velocity less than a lighter ion. Mass and charge therefore enter the conditions for successful motion together.

The drive combines a constant DC voltage with a rapidly alternating radio-frequency voltage. DC leaves a different average tendency along the two transverse directions. Adjusting the DC-to-RF ratio can narrow the range that remains bounded in both directions. RF alone usually acts more like a broad high-pass filter in the usual first operating region.

In the usual operating region, very light ions can develop large RF-driven excursions. Very heavy ions can be lost along the direction that DC defocuses because the RF drive no longer stabilizes them sufficiently. An intermediate range can meet both transverse conditions. Changing the voltage ratio changes the width of this range.

Sources: [3]

## The stability map imagines an unlimited flight

The ideal quadrupole field leads to the Mathieu equation, which describes an oscillator with a periodically changing force. Two dimensionless parameters combine the drive and ion inertia. Some settings give motion that stays bounded indefinitely; other settings give growing excursions. A mass filter needs both transverse directions to satisfy the bounded-motion condition.

Sources: [5]

## Finite rods change the measured transmission

A real ion has only a finite distance to travel. A trajectory that would eventually grow into a rod can leave a short filter first. Conversely, bounded motion can still hit metal if the particle starts too far off axis or enters at too large an angle. Beam width and transit time affect the collected signal.

Narrowing the pass band generally separates nearby mass-to-charge ratios more selectively and transmits fewer ions. The number of RF cycles experienced during flight also limits the separation. Short rods or fast ions provide fewer opportunities for an unwanted trajectory to grow. Selectivity and throughput must be considered together.

Sources: [4]

## Known ions set the scale

The collector measures current, or an amplified detector provides ion counts. A voltage scan becomes a calibrated mass scale when ions of known mass-to-charge ratio locate its peaks. Another known species can check that scale independently and reveal a tuning shift. Different ions can share the same mass-to-charge ratio; identifying a particular molecule also draws on fragmentation and other chemical information.

Sources: [2] [4]

## [Compare rod impacts with collector counts](https://sandboxphysics.com/labs/quadrupole-mass-filter/)

Shorten the rods and compare stability with finite transmission

Integrates finite flights in an ideal hyperbolic field and calibrates the count spectrum with independent known ions.

## Sources and further reading

1. [Wolfgang Paul · Nobel Lecture · 8 December1989](https://www.nobelprize.org/uploads/2018/06/paul-lecture.pdf)

   Selected text around printed pp603–605, equations1–6 and the mass-filter stability discussion; pair-to-pair voltage convention. Full lecture reading is not claimed.

2. [Agilent7200 Q-TOF Concepts Guide · G3850-90003 · February2012](https://www.agilent.com/cs/library/usermanuals/public/G3850-90003.pdf)

   Read printed pp32–34 on single-quadrupole operation, scanning, selected-ion monitoring and the source/filter/collector chain.

3. [Pfeiffer Vacuum · Knowledge book6.3.1](https://www.pfeiffervacuum.com/no/en/knowledge/vacuum-technology/knowledge-book/6-mass-spectrometers-and-residual-gas-analysis/6.3-quadrupole-mass-spectrometers-%28qms%29.html)

   Read6.3.1, formulas6-5–6-12 and injection/geometry/RF-only discussion. Exact stability references use independent characteristic values.

4. [Stanford Research Systems QMS100 Manual · Version3.2 · January2012](https://www.thinksrs.com/downloads/pdfs/manuals/QMSm.pdf)

   Read technical-reference pp2-15–2-17: finite cycles, selectivity/throughput, zero blast and detection. The manual defines width at10% peak height.

5. [NIST DLMF · Mathieu equation and stability](https://dlmf.nist.gov/28.17)

   Read28.17 stability definition and characteristic boundaries;28.2 equation and fundamental-matrix normalization.

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