Quadrupole Mass Filter

A quadrupole mass filter (QMF) is a key component of a modern mass spectrometer. A QMF uses direct current (DC) and alternating current (AC) electric fields to analyze positive or negative ions by mass-to-charge ratio. A QMF consists of four parallel rods spaced equidistantly. The ratio of the rod radius to the radius of the inscribed circle is 1.148 (Ref. 1). Opposite pairs of rods are electrically connected. Adjacent rods have opposite DC potentials and their AC potentials are out of phase. Typical rod diameters are between 5 and 12 mm with rod lengths between 100 and 200 mm. The frequency of the AC component of the electric field is typically in the range 1 to 10 MHz.

Ions are created using a number of different techniques depending on the application. They are injected along the axis of the QMF. Typical ion energies at the QMF entrance aperture are 3 to 5 electron volts. The ions experience forces due to the AC and DC fields near the quadrupole axis. For certain AC and DC field values only an ion of a specific mass to charge ratio is transmitted. Ions are detected by measuring the ion current at the exit of the quadrupole.

This application requires the Particle Tracing Module.

Model Definition

The cross section of a typical QMF is shown in Figure 1. The ions are injected in such a way that their initial velocity is directed only in the out-of-plane direction. As long as the ions remain in the QMF they experience forces in the x and y direction due to the DC and AC fields. The rod radius is 2.78 mm and the radius of the inscribed circle is about 2.42 mm. Since this is a 2D model, there is no rod length. However, since the ion velocity is assumed constant in the out-of-plane direction, the effective rod length can be computed by multiplying the out-of-plane ion velocity by the total simulation time. The total simulation time is 40 μs, which, for an RF frequency of 4 MHz, corresponds to 160 RF cycles. For an initial ion energy of 3 eV and an ion mass of 40 amu, this corresponds to an effective rod length of 152 mm.

Figure 1: Plot of the model geometry. The geometry length unit is meters.

This application is designed to find the transmission probability of ions in the QMF for different values of two dimensionless parameters, a and q. These are coefficients in the Mathieu equation, which can be used to solve the same problem as this application. However, this application can be generalized to 3D and the effects of fringing fields may also be included. The parameters a and q are scaled values for the DC and AC voltages respectively. It is important to note that in order to gain an accurate statistical measure of the transmission probability, it may be necessary to solve for a much larger number of particles (Ref. 1). The parameter, a, is related to the applied DC voltage by:

where

•

Z (dimensionless) is the charge number,

•

e = 1.602176565 × 10-19 C is the elementary charge,

•

Udc (SI unit: V) is the applied DC voltage,

•

m (SI unit: kg) is the particle mass,

•

r0 (SI unit: m) is the inscribed radius, and

•

ω (SI unit: Hz) is the angular frequency.

The parameter q is related to the applied AC voltage by:

By solving the Mathieu equation it is possible to construct a stability diagram that shows whether the particles undergo stable or unstable oscillatory motion down the QMF. The stability diagram is plotted in Figure 2.

Figure 2: Stability diagram for a quadrupole mass filter.

So long as the values of a and q (and thus Udc and Vac) remain within the gray region, the particles do not make contact with the rods. The operating principle of a QMF is to sweep through a range of values for q whilst keeping the ratio of a/q constant. The idea is illustrated in Figure 3. The values of a and q are both increased simultaneously such that they follow the red line. The ion trajectories are initially unstable, which results in a transmission probability of zero. Once the value of q reaches about 0.6, the ions enter the stable operating region and some particles are transmitted to the detector. As the value of q increases further, the ions end up back in the unstable region and the transmission probability is reduced to zero.

Figure 3: Operating principle of a quadrupole mass filter.

For the DC field, the application solves Poisson’s equation for the electric potential, U:

(1)

Here ε0 is the permittivity of free space (SI unit: F/m) and εr is the relative permittivity of the medium (taken as 1 in this application). The zero on the right-hand side of Equation 1 indicates that the space charge density inside the quadrupole is negligible. On the north and south rods, a positive potential of magnitude Udc is applied:

and on the east and west rods, a negative potential is applied:

For the AC fields, the conservation of electric currents is used to compute the AC potential, V:

where σ is the electrical conductivity in the filter (taken as zero here) and ω is the angular frequency (SI unit: Hz).

On the north and south rods, a positive potential of magnitude Vac is applied:

and on the east and west rods, a negative potential is applied:

In order to construct the total electric field which the particles experience once they enter the modeling domain, superposition of the AC and DC fields is used. This is a valid assumption in this case since the equations solved for the AC and DC fields are linear.

Particle motion is governed by Newton’s second law:

where m is the particle mass (SI unit: kg), v is the particle velocity (SI unit: m/s), Z is the dimensionless charge number, e is the elementary charge (SI unit: s A), and E is the electric field (SI unit: V/m). The electric field contains two contributions, a stationary electric field and one which is changing over time:

where:

and

where the tilde denotes that the AC electric potential is complex-valued. The particle position, q, is simply computed from the definition of the velocity:

The particles have no initial velocity in the modeling plane. Particles are not only released at the simulation start time, they must be released at uniformly spaced times over the first RF cycle of the AC field. Particles are released at 11 times from 0 s to 0.25 μs; since the frequency of the AC field is 4 MHz, this corresponds to one full RF cycle.

Results and Discussion

The location of the ions for different values of the parameter q after 140 RF cycles are plotted in Figure 4. As expected, all the injected ions make contact with the rods up to a value of q = 0.6. For q = 0.7, however, there are ions that have clearly not made contact with the rods. These ions are detected by a current collector at the end of the QMF.

Figure 4: Plot of the ion location after 140 RF cycles inside the QMF.

The transmission probability is plotted in Figure 5. There is a clear stable window of operation between the values q = 0.6 and q = 0.75. This result agrees well with the theory outlined in the Model Definition section. In this example the slope of a/q was fixed to be 0.2/0.7, meaning the width of the transmission peak spans over a q range of 0.15. If the slope of a/q were to be increased, the width of the peak where stable operation occurs would be reduced. If the slope of the curve is greater than 0.237/0.706 then there are no regions of stable operation for any value of q.

As the slope of a/q decreases, the range of q values for which the ion trajectories are stable increases.

Figure 5: Plot of the transmission probability through the QMF for different values of q.

Notes About the COMSOL Implementation

Use the feature on the parameter q while keeping the ratio a/q constant. For each value of q, separate study steps are called to solve for the fields and particle trajectories.

Reference

1. J.R. Gibson, S. Taylor, and J.H. Leck, “Detailed simulation of mass spectra for quadrupole mass spectrometer systems,” J. Vac. Sci. Technol. A, vol. 18, no. 1, p. 237, 2000.

ACDC_Module/Particle_Tracing/quadrupole_mass_filter

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

In the tree, select .

Click .

In the tree, select .

Click .

Click

Global Definitions

Add some parameters for the quadrupole geometry and physics settings. To save time, the parameters can be loaded from a file. The optimum inscribed radius is the rod radius divided by 1.147.

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file quadrupole_mass_filter_parameters.txt.

The parameter q will be varied during a but the ratio of the DC and AC applied voltages will stay the same. The ionic mass is that of argon.

Geometry 1

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type rcase.

Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type re.

Locate the section. In the text field, type re+r0.

Circle 3 (c3)

In the toolbar, click

In the window for , locate the section.

In the text field, type re.

Locate the section. In the text field, type re+r0.

Circle 4 (c4)

In the toolbar, click

In the window for , locate the section.

In the text field, type re.

Locate the section. In the text field, type -(re+r0).

Circle 5 (c5)

In the toolbar, click

In the window for , locate the section.

In the text field, type re.

Locate the section. In the text field, type -(re+r0).

Circle 6 (c6)

In the toolbar, click

In the window for , locate the section.

In the text field, type rsrc.

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Select the

toggle button.

Select the objects , , , and only.

Click

. The geometry should look like Figure 1.

Materials

The ion trap is at near vacuum conditions. The relative permittivity is hence 1 and the conductivity 0.

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic

Definitions

It is convenient to define some selections for the positive and negative electrodes.

Positive

In the toolbar, click

In the window for , type Positive in the text field.

Locate the section. From the list, choose .

Select Boundaries 7–10, 14, 15, 18, and 19 only.

Negative

In the toolbar, click

In the window for , type Negative in the text field.

Locate the section. From the list, choose .

Select Boundaries 3–6 and 21–24 only.

Set the to so that the correct equation contribution is generated. This is necessary because the study step will be used to compute both the AC and DC fields later on.

Electrostatics (es)

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Change the name of the dependent variable for the DC field to U.

Click to expand the section. In the text field, type U.

Now define the boundary conditions for the DC field.

Electric Potential 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the V0 text field, type Udc.

Electric Potential 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the V0 text field, type -Udc.

Electric Currents (ec)

In the window, under click .

In the window for , click to expand the section.

In the text field, type V.

Now define the boundary conditions for the AC field.

Electric Potential 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the V0 text field, type Vac.

Electric Potential 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the V0 text field, type -Vac.

Specify the properties of the ions. The charge number remains at 1 and the ion mass is set by a parameter which was defined eariler.

Charged Particle Tracing (cpt)

Particle Properties 1

In the window, under click .

In the window for , locate the section.

In the Z text field, type 1.

Locate the section. In the mp text field, type mi.

In the window, click .

In the window for , locate the section.

In the text field, type 0.

The particles are released based on the mesh elements in the aperture. Furthermore, the particles are released at different phases of the first RF cycle.

Release 1

In the toolbar, click

and choose .

Select Domain 2 only.

In the window for , locate the section.

Click

In the dialog box, choose from the list.

In the text field, type 0.

In the text field, type 1/4E6.

In the text field, type 11.

Click .

There are two forces acting on the particles. One is due to the DC electric field and one is due to the AC electric field. To add these, simply add two features. The check box is selected so that the electric field is as accurate as possible.

Electric Force 1

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select both domains.

In the window for , locate the section.

From the E list, choose .

Locate the section. Select the check box.

For the AC force, the check box must be selected. This ensures that the magnitude of the electric field is multiplied by ejωt as the particles are traveling down the trap.

Electric Force 2

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select both domains.

In the window for , locate the section.

From the E list, choose .

Locate the section. From the list, choose .

Select the check box.

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Click

Now add the study to compute the fields and the particle trajectories. Solve the problem using two study steps. First, compute the AC and DC electric fields. Then, using the computed fields to define the electric force, solve for the particle trajectories. These steps are wrapped with a parametric sweep over the parameter q. The number of parameters in the sweep is increased between the limits of 0.6 and 0.8 since this is where the transmission probability will be highest.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Study 1

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
q (Mathieu coefficient)	0.1 0.5 range(0.6,0.2/10,0.8) 1

Frequency Domain

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 4E6.

Locate the section. In the table, clear the check box for .

Time Dependent

In the toolbar, click

and choose .

In the window for , locate the section.

Click

In the dialog box, type 160/4e6 in the text field.

In the text field, type 1/4e6.

Click .

In the window for , locate the section.

In the table, clear the check boxes for and .

Click to expand the section. Find the subsection. From the list, choose .

From the list, choose .

In the toolbar, click

Results

Three plots are created by default. The first two show the electric potential distributions for the AC and DC fields.

Particle Trajectories (cpt)

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

The particle positions at the final time are plotted for q = 0.1. To reproduce the remaining plots in Figure 4, select the values , , and from the , then click .

Particle 1

In order to compute the transmission probability, it is necessary to add a second dataset, and then add a to it. The number of particles counted on this selection divided by the total number of particles is the transmission probability for the selection. This value is stored in a variable called alpha. The fraction of particles that do not strike the rods, 1-alpha, is the transmission probability for the QMF.

Particle 2

In the window, expand the node.

Right-click and choose .

Selection

In the toolbar, click