# 4.1.2.1 Quadrupole mass filter

The filter system of a quadrupole mass spectrometer consists of four parallel rods arranged in the form of a square. Each pair of opposite rods in Figure 4.4, designated (+) or (−), is connected with the other. Voltage

#### Formula 4-1:

Quadrupole deflection voltage

is applied between the two pairs of rods.

At this point, only a brief phenomenological description of the operating principle will be provided. Reference is made to the literature for a detailed presentation [14, 15, 16].

**Figure 4.4:**
Operating principle of a quadrupole mass spectrometer

Ideal quadrupole fields require rods that have a hyperbolic profile. In actual practice, however,
round rods are used, with the rod radius being equal to 1.144 times the field radius `r _{0}`.
An electrical quadrupole field is formed between the rods. Ions of varying mass are shot
axially into the rod system at approximately equal energy and move through the rod system
at uniform velocity. The applied quadrupole field deflects the ions in the X and Y directions,
causing them to describe helical trajectories through the mass filter. To solve the movement
equations, the dimensionless variables

#### Formula 4-2:

Stability parameter a

and

#### Formula 4-3:

Stability parameter q

are introduced to obtain Mathieu's differential equations. Their solutions yield the stable area
with oscillation amplitudes of less than `r _{0}` beneath the triangle formed by the two solubility
curves in Figure 4.5. The values

`a`= 0.23699 and

_{p}`q`= 0.706 apply for the apex of the triangle. All solutions outside result in increasing oscillation amplitudes and thus in neutralization of the ions on the rods of the quadrupole filter.

_{p}Dividing the two equations by one another yields: .

This is the pitch of the so-called load line of the mass filter.

**Figure 4.5:**
Stability diagram of a quadrupole filter

From Figure 4.5, it can be seen that:

- All ions whose parameters a and q are located in the triangle above the load lines will reach the detector
- The ions will only reach the detector under voltage conditions < 0,1678.

Introducing the ratio between the atomic mass unit 1 amu = 1.66055 · 10^{-27} kg and the
elementary charge and multiplying it by
the dimensionless mass number `M` of the corresponding ion yields the following conditions
for `U` and `V` for the apex of the stability triangle:

#### Formula 4-4:

Stability condition `U`

and

#### Formula 4-5:

Stability condition `V`

With the DC voltage de-energized, `U` = 0, all trajectories of the ions where `q` < 0.905
will be stable; according to Formula 4-5, these will all be masses where

#### Formula 4-6:

High-pass condition

The filter thus acts as a high pass. As HF amplitude (`V`) increases, ever-heavier types of ions
become unstable, starting with the light masses, and are thus sorted out. This operating
mode produces an integral spectrum

The shot conditions are crucial for transmission of ions through the filter. Ions parallel to the rod system must be shot in within the following diameter

#### Formula 4-7:

Shot orifice

The maximum shot angle must satisfy the condition:

#### Formula 4-8:

Shot angle

and the energy must be as uniform as possible. The advantages of the Pfeiffer Vacuum ion sources described in 4.1.4.1 translate into high transparency and thus high sensitivity.

In order for the amplitudes of the unstable ions to become large enough to strike the rods, where they are neutralized, these ions must perform a minimum number of oscillations in the separating field. The following equation applies for the maximum acceleration voltage in the Z direction:

#### Formula 4-9:

Maximum acceleration voltage U`zmax`

In practical operation, the ratio `U/V` is activated as a function of the mass number in such a
manner that the actual resolution `ΔM/M`, does not remain constant, but that instead the line
width `ΔM` remains constant. This means that resolution increases proportionally to the mass
number. Due to Formula 4-5 (`V` is proportional to `M`), the quadrupole
(as opposed to the sector field mass spectrometer) produces a linear mass scale.

One point of major significance for a QMS is the required HF power. If `C` is used to designate
the entire capacity of the system and `Q` to designate the factor quality of the power circuit,
the required HF power

#### Formula 4-10:

HF power

will increase with high powers of f and r0. An enlargement of field radius `r _{0}` will
lessen the occurring relative mechanical tolerances, thus resulting in improved behavior. Essentially, it is
advantageous to select

`f`and

_{0}`r`as large as possible. However there are limits in this regard due to the associated increase in HF power (Formula 4-10). While extending the rod system permits a lower operating frequency, the size of a production unit should not exceed certain dimensions.

_{0}The required mass range and desired resolution are governed by the dimensions of the filter and the selection of the operating frequency. Devices with 6, 8 and 16 mm rod diameters and appropriately matched electronics are available to satisfy the respective requirements.

What follows is a brief digression on the relationship between resolution and mechanical precision. Let us consider a quadrupole mass filter that works at the apex of the stability diagram; i.e. that works at high resolution.

applies for the DC amplitude and

for the AC amplitude.

Here, `M` designates the mass of the ion, `r _{0}` the field radius and

`f`the frequency at which the filter is operated. We are making the idealized assumption that both voltages

`U`and

`V`, as well as frequency

`f`, can be set and maintained “as precisely as desired”.

What follows from this is:
(`c _{k}` is a constant), and following differentiation and division by

`M`, the filter scatter caused by

`r`is:

_{0}#### Formula 4-11:

Scatter

Let us assume that the field radius `r _{0}` changes by

`Δr`= 0.03 mm over the length of the mass filter. Now let us consider the effect of this change on scatter

_{0}`ΔM/M`for two mass filters of different sizes. For optimal transmission, the resolution set on the spectrometer (we select:

`ΔM/M`= 1/100) must be greater than the scatter generated by the fluctuation of

`r`:

_{0}.

For a filter (a) having a field radius of 3 mm, this results in `ΔM/M` = 0.02, i.e. a
contradiction; and for a filter (b) having a field radius of 12 mm, this results in `ΔM/M` = 0.005,
i.e. coincidence with the above condition. In other words: If a resolution of `ΔM/M` = 0.01 has been
set for both filters, most of the ions will not be able to pass through the filter in case (a). In
the case of the larger filter (b), all ions will be able to pass through the filter, since the set
resolution is greater than the scatter.

Although this simplified error calculation does not nearly take into account all of the effects that can contribute to transmission, it does teach several fundamental relationships:

- The field radius must be maintained significantly better than 1% over the entire length of the filter, depending on the selected mass range. Fluctuations in the field radius will lead to transmission losses
- The larger the dimensions of the rod system are selected, the lower the influence of the absolute mechanical tolerances will be
- The higher the mass range, and if differentiation between adjacent masses is still to be made, the stricter will be the requirements relating to the relative accuracy of the mass filter. In this simple picture, the required relative accuracy is scaled linearly with the mass range

## Summary

A quadrupole mass filter is a dynamic mass filter for positive and negative ions. The mass
scale is linear to the applied amplitude of the HF voltage. Mass resolution can be conveniently
and electrically set by means of the ratio between the DC voltage `U` and the high-frequency
voltage amplitude `V`. Due to their small dimensions and light weight, quadrupole mass
spectrometers are suitable both as pure residual gas analyzers and, in higher-quality design,
as sensors for gas analysis.

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