Original research paper
Numerical analysis of trajectories in a Cassinian
ion trap of second order with trap door ion inlet
Bjoern Raupers1, Hana Medhat2, Juergen Grotemeyer1
Frank Gunzer2
European Journal of Mass Spectrometry
2021, Vol. 27(1) 3–12
! The Author(s) 2020
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DOI: 10.1177/1469066720984380
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Abstract
Ion traps like the Orbitrap are well known mass analyzers with very high resolving power. This resolving power is achieved
with help of ions orbiting around an inner electrode for long time, in general up to a few seconds, since the mass signal is
obtained by calculating the Fourier Transform of the induced signal caused by the ion motion. A similar principle is applied
in the Cassinian Ion Trap of second order, where the ions move in a periodic pattern in-between two inner electrodes. The
Cassinian ion trap has the potential to offer mass resolving power comparable to the Orbitrap with advantages regarding
the experimental implementation. In this paper we have investigated the details of the ion motion analyzing experimental
data and the results of different numerical methods, with focus on increasing the resolving power by increasing the
oscillation frequency for ions in a high field ion trap. In this context the influence of the trap door, a tunnel through
which the ions are injected into the trap, on the ion velocity becomes especially important.
Keywords
Cassinian ion trap, ion traps, electrostatic ion traps, high field ion traps, mass spectrometry
Received 14 October 2020; accepted 8 December 2020
Introduction
Mass analyzers are very helpful instruments in a variety of applications.1–3 These include identification of
substances and structure elucidation. In the latter case
for example, the detection of many masses of fragments formed under different circumstances from
the original molecule is necessary. The more precise
the mass can be determined, especially regarding fractions of unit mass, the easier it is to identify the
molecular structure of a complete molecule or fragments since the detailed chemical composition
becomes more and more the decisive factor for a certain mass value. The quantity that describes this precision is the mass resolving power, which is basically
the ratio of the width of a mass signal and its location
(Dm/m). A resolving power of 10,000 is often sufficient for the unambiguous identification of analytes
with masses smaller than 100 dalton. To be precise,
this is valid only if the mass accuracy is also good
enough, and if the analyte charge is one, since mass
analyzers typically yield the mass to charge ratio, and
not the mass alone. A resolving power of 1,00,000
correspondingly extends this range. One mass analyzer with resolving power in this range, from hundreds
of thousands to over a million, with relatively modest
experimental requirements is the Orbitrap.4–6 Here,
specially shaped outer and inner electrodes create an
electric field which forces injected ions onto an orbital
motion around the inner electrode. This orbital
motion furthermore oscillates into a certain direction
typically called the Z axis of the motion, which is then
the longest axis of the trap. The specific form of the
electric field yields a Z motion with a frequency
which, among other factors, is also depending on
the analyte’s mass to charge ratio. Measuring this frequency then allows for the determination of the analyte mass. The ion motion induces a signal in the
outer electrodes, and the Fourier Transformation
(FT, most often in its computationally more efficient
form of Fast Fourier Transformation FFT) of this
signal yields the required frequency. Since the peak
width of a FFT signal scales in a reciprocal fashion
1
Department of Laser Mass Spectrometry, Institute for Physical Chemistry,
Christian-Albrecht-University Kiel, Kiel, Germany
2
Department for Electronics Engineering/Center for Computational
Engineering, Faculty of Information Engineering and Technology, German
University in Cairo, Cairo, Egypt
Corresponding author:
Frank Gunzer, Department for Electronics Engineering/Center for
Computational Engineering, Faculty of Information Engineering and
Technology, German University in Cairo, Entrance El Tagamoa El Khames,
New Cairo City, Cairo, Egypt.
Email: frank.gunzer@guc.edu.eg
4
with the signal duration in the time domain, measuring for longer times increases consequently the precision of the analyte mass calculation. This is the
strength of such ion traps, which allow for long measurement times of up to a few seconds and consequently for very high mass resolving powers. The
Orbitrap is a central part of a great number of scientific as well as industrial discoveries and break
through applications.
The orbital motion in these analyzers can only be
achieved if the injection conditions are carefully
chosen and controlled. Examples are the ions’ radial
position which needs to be maintained at values
where the electric force attracts the ions towards the
center, negligible initial radial velocity component
during injection, or the injection energy which if
reaching certain values might lead to elliptical instead
of orbital trajectories. One relatively young set up
that works with the same principles but a slightly different detail and therefore not necessarily requiring
any form of injection is the Cassinian Ion Trap of
Second Order.7–10 Second order means here that
there are two inner electrodes (even higher orders
are possible); the shape of the electrodes have been
calculated with help of Cassinian Curves which give
the trap its name. This trap allows for a number of
ion trajectories (called modes) of different and especially non-orbital shapes. One very interesting mode
forms if the initial ion position is located in-between
the inner electrodes at a location with Z not equal
zero (i.e. not directly in the center of the trap).
Then a stable ion trajectory with an oscillation into
Z direction will automatically form even for a particle
without any initial kinetic energy, i.e. initially at rest.
For other initial velocity vectors, a stable trajectory
can form as long as the initial velocities are within
certain limits. The initial position regarding X-, Yand Z-coordinates as well as the initial velocity vectors determine the precise shape of the trajectory. If,
as indicated before, the initial position is in-between
the inner electrodes regarding the X-axis (which is the
distance vector between the inner electrodes, see
Figure 1) and the initial velocity vectors are within
certain limits, the ion trajectory remains within a certain cuboid-shaped volume so that it does not collide
with the inner electrodes nor with the outer electrode;
theoretically the resulting trapping motion can extend
to infinity in time, and practically to very long times if
losses can be reduced to a minimum. These losses are
here especially caused by collisions with remaining
gas atoms, so ion traps have to be operated in
vacuum. The performance is then comparable to
that of an Orbitrap since the signal detection follows
the same principles. Further advantages apart from
the injection method, are that the vacuum is easier
maintained inside the trap since there are more openings (see e.g. Makarov et al.11 for a discussion of the
importance of that point in case of the Orbitrap).
European Journal of Mass Spectrometry 27(1)
Figure 1. Geometry of the Cassinian Ion Trap of second order.
The trap center defines the location (X,Y,Z) ¼ (0,0,0). The left
image shows a cut along the Z-Y-plane of the trap. The Z-axis is
the longest axis of the trap, the Y-axis is in that image the vertical
axis. The right image shows the arrangement of the two inner
electrodes (X-axis is from left to right, Y-axis is again the vertical
axis). The small tunnel in both images through the outer electrode along the Y-axis is the ion inlet, here called trap door. The
ions when injected through the trap door at X ¼ 0 with suitable
initial velocity only in y-direction will fly in-between the inner
electrodes along the Z-axis in a Z-Y-plane at X ¼ 0.
Although not strictly required for a Cassinian Trap
of Second Order, ions normally are also here collected
outside this trap (e.g. by another trap principle such
as a Paul Trap) and then injected into it. A simple and
convenient way to achieve that is to have a small
tunnel leading through the outer electrode. This
tunnel is called trap door.7,8 For positively charged
ions the trap door is connected to a certain negative
electric potential during injection. Once the ions are
inside the trap, the trap door is connected to ground.
The outer electrode is permanently connected to
ground, and the inner electrode permanently connected to a negative potential. This potential determines the resolving power, so that not having to
switch it like in e.g. Orbitraps means benefits for the
resolving power; the inner electrodes can be connected to very stable voltage supplies such as batteries. Switching voltages with power supplies typically
introduces fluctuations also regarding the finally
reached voltage, which is here supposed to be constant. These fluctuations would then directly affect
the resolving power, but batteries can be much
more stable in that regard. The negative trap door
potential can be chosen in relation to the initial kinetic energy of the ions. If the ions are accelerated by the
trap door potential while coming from a certain negative potential which is higher than that of the trap
door, they will travel through the trap door and automatically be trapped inside the ion trap. Once the trap
door potential is switched back to ground they cannot
reach the trap door anymore, since initially coming
from a negative potential they cannot reach any point
on ground potential if the initial velocity at that start
point is slow enough. Typically, the trap door is constructed perpendicular to the central plane formed by
the inner electrodes, so that after injection the ions
have only initial velocity in that direction (here called
Y direction, see Figure 1).
Raupers et al.
5
Mathematically, the ion trajectories can be described in the following way. The electric potential that forms inbetween the outer and inner electrode follows this formula:7,8
0
Bln
B
B
Wðx; y; zÞ ¼ B
@
x2 þ y2
2
1
!
2 b2 x2 þ y2 þ b4
C
F ð1 BÞ x2 B y2 þ z2 ai2 ð1 FÞC
ai4
C
þ
C
A
ao2 ai2
ao4
ln
4
ai
ðUo UiÞ þ Ui
This potential includes the parameters outer electrode diameter ao, inner electrode diameter ai, fraction of
logarithmic potential F, focal point of the Cassinian Curves b, additional geometric stretching constant in Ydirection B, outer electrode potential Uo and inner electrode potential Ui. From this equation, the Newtonian
Equations of motion can be obtained.
0
1
d
4 x2 þ y2 x 4b2 x F
2ð1 BÞxð1 FÞ
m 2 x ¼ q @
AðUo UiÞ
4
dt
ao2 ai2
ðx2 þ y2 Þ2 2b2 ðx2 y2 Þ þ b4 ln ao
ai4
0
1
d
4 x2 þ y2 y 4b2 y F
2Byð1 FÞ
m 2 y ¼ q @
AðUo UiÞ
4
dt
ao2 ai2
ðx2 þ y2 Þ2 2b2 ðx2 y2 Þ þ b4 ln ao
ai4
2ð1 FÞðUo UiÞ
¼ q z ¼ q C z
2 ai2
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ao1
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
rffiffiffiffiffiffi
q
m
q
! zðtÞ ¼ z0 cos@
C tA þ v0;z sin@
C tA
m
qC
m
m
d
z
dt2
For the Z coordinate, the solution is very simple and yields the harmonic motion with a frequency that depends
on the analyte’s mass to charge ratio which can be determined via the induced signal on the outer electrode (if
proper initial conditions are chosen). The motion in Z direction is independent from the motion in the other
directions. For a motion that begins at X ¼ 0 in-between the inner electrodes, the motion remains in the Y-Z-plane
at X ¼ 0. This motion is much simpler than the orbital motion in an Orbitrap which is a motion in the X-Y-plane
that oscillates in Z-direction. The motion in the other directions is more complicated. Typically, these motions dephase more quickly than the one in Z-direction. The final ion cloud shape in the Cassinian Trap then resembles
that of a band of ions in Y-direction that oscillates in Z-direction. After relatively short time the induced signal is
therefore almost exclusively created by the motion in Z-direction with a correspondingly simple FFT frequency
spectrum.
The induced signal can be calculated with help of the Shockley-Ramo-Theorem.12,13 It can be derived from
Green’s Reciprocity Theorem and described mathematically in this form:
QA ¼ qV
dQA
@V dr
¼ q
¼ q E v
iA ¼
dt
@r dt
The charge q induces the charge QA on the electrode A. The potential V* is the so called weighting potential.
This is a potential calculated by setting all electrodes in the set up to zero potential (ground), and only the
electrode A, for which the induced charge is to be calculated, to a potential of one volt. In general, this is not
the potential that moves the charge. Since the induced current is related to the change of charge per time, the time
derivative of this charge leads to the formula iA¼qE*v. The quantities r, E* and v given in the formulas are to be
interpreted as vector quantities.
In this paper, we have analyzed the resulting ion trajectories with help of finite elements method simulations
(FEM), as well as with help of numerical solutions of the Newtonian Equations of motion. The goal was to reach
a geometry with increased resolving power. The Newtonian Equations of Motion do not include the disturbance
of the field caused by the trap door, so that the FEM simulations were used to calculate the trajectories for a
realistic set up. On the other hand, the FEM simulations take quite a long time, which makes it difficult to derive
6
the basic principles of the motion from the FEM simulations and e.g. find suitable initial conditions.
Therefore, the numerical solutions of the Newtonian
Equations of Motion were used to derive basic principles that can be helpful to optimize the geometry.
The result is that with slight changes of the geometry
and increased voltage, it was possible to increase the
frequency of the Z-motion by 60.5% and therefore
the resolving power by this factor assuming that all
other factors influencing the resolving power remain
the same. On the other hand the numerical analysis
shows that when using a trap door in the same fashion as described here, a further significant increase of
frequency/resolving power seems very difficult.
It should be stressed here that obtaining stable ion
trajectories for a changed trap geometry and
increased voltage difference as required to increase
the resolving power is by no means trivial. After a
short descriptions of the methods used in our analysis, this paper will first describe how well FEM is able
to reproduce characteristic values (e.g. frequencies)
obtained with an experimental set up. Following
that part we will illustrate how with help of the
Newtonian Equations of Motion we could derive stability criteria for the ion motion, and based on these
criteria we were able to investigate geometric changes
of a realistic set up leading to a better resolving
power. These could then be verified by FEM
simulations.
Experimental
In the FEM simulations, the following parameters
have initially been used in the trap potential formula:
Outer electrode diameter ao 12 mm, inner electrode
diameter ai 6.5 mm, fraction of logarithmic potential
F 0.9, focal point b of the Cassinian Curves is 7 mm,
stretching constant B 1.75. The trap door is a tunnel
through the outer electrode along the Y-axis with
diameter 0.8 mm. It is electrically insulated from the
outer electrode so that its potential can be chosen
independently. The same geometric parameters have
been used in the experimental set up7,8 so that for this
set up certain simulation results could be compared to
experimental results since we could employ an identical device for our measurements. The shape of the
outer and inner electrodes is obtained by setting the
outer and inner electrodes to the desired voltages
(meaning here a difference of 4000 V between outer
and inner electrode) and solving the trap potential
formula for the corresponding coordinates. Figure 1
shows the obtained shapes.
In the FEM simulations, the Laplace Equation has
been solved in-between outer and inner electrode. The
mesh element size was 0.0001 mm near the electrodes,
and reached a maximum value of 0.4 mm. The total
number of mesh elements was ca. 3000000. So called
quadratic elements were chosen which is necessary to
be able to calculate derivatives of the potential field;
European Journal of Mass Spectrometry 27(1)
the correlated functions also fit well to the mathematical form of the Cassinian Potential which depends on
the coordinates to a large extent in a quadratic
manner. It was verified that the mesh size does not
influence the obtained field (deviations for different
mesh sizes below 1 mV). The ion trajectories were
simulated with help of charged particle tracing. The
number of particles was 2000, each with one positive
charge and a mass of 479.08 Dalton; this is the mass
of rhodamin B, for which experimental spectra for
this set up were available. The in the case discussed
here positively charged ions enter the trap through the
trap door, which was set to a voltage of 800 V with
the ions starting at a potential of 150 V (similar to
Koester7,8) After entering the trap, the trapdoor is
switched back to ground. This ensures that the trap
door does not disturb the ion motion, and furthermore that the ions cannot leave the trap since they
cannot reach a potential of zero volt since they started
at 150 V. The inner electrodes are set to 4000 V
and the outer electrode remains at ground. This
does not change the ion motion although the
Cassini potential has been calculated for 4000 V on
the outer electrode and 0 V on the inner electrodes,
since the potential difference is the same. All these
parameters are similar to those used for recording
the experimental spectra. In order to achieve the necessary precision allowing for a simulation time of up
to 3 ms, a time step magnitude of 1 ns had to be
chosen, as was found out in preliminary simulations.
Due to memory restrictions, it was not possible to
simulate for longer times. Coulombic effects and collisions with background gas were excluded. Typical
trajectory shapes can be found e.g. in Raupers et al.10
The induced signal was calculated by using the
Shockley-Ramo-Theorem, the necessary weighting
fields were calculated by solving the Laplace equations with all electrodes set to ground, except for
the outer electrode. The outer electrode is in real
experiments split into two halves and the induced
signal measured with help of a differential amplifier.
Correspondingly, the weighting fields were here calculated for one half of the outer electrode set to one
volt, and the other half to ground, and vice versa. The
final induced signal was then also calculated as
the difference of the induced signals in each half.
The mass spectra were finally obtained by using
FFT on that induced signal.
The experimental spectrum was measured with a
set up identical to the one described in7,8 including the
Paul-Trap. The Cassinian trap has the same geometric details as described before for the simulations
which were actually based on the existing device. As
analytes, rhodamine B (Radiant Dyes Laser &
Accessories GmbH, Germany) and reserpine (Sigma
Aldrich, Germany) were used. The samples were first
dissolved in ethanol respectively methanol, and then
they were further diluted in a mixture of methanol
and water (1:1) with 0.2% formiatic acid to a final
Raupers et al.
concentration of 10 mM. Finally, the analytes were
sprayed in electrospray ionization (ESIþ) mode.
The pressure inside the Cassinian trap was 1010
mbar. The time step size of the measured signal was
1.07 ls, total measurement time was 3,07,800 time
steps (ca. 328 ms).
Stability diagrams for the ion motion were calculated by solving the differential equations numerically
with a 4th order Runge-Kutta approach.
Results and discussion
This section contains two parts. In the first part, it
will be shown how well FEM simulations of the original set up are able to reproduce quantities that can be
obtained with a similar, existing experimental set up.
In the second part, numerical solutions of the
Newtonian Equations of Motion will be used in
order to adjust the trap geometry and ion injection
voltages so that the resolving power can be increased.
These Equations of Motions are based on a Cassinian
Field with the same mathematical description as the
one present in an ideal Cassinian trap. Therefore, it is
not possible to e.g. know when a trajectory would
collide with an inner or outer electrode, and similarly
the influence of the trap door on the field is not
included. But these Equations of Motion can help
to find suitable initial conditions that lead to ion trajectories that in the ideal trap do not collide with the
inner or outer electrodes. The more realistic trajectories including collisions with the electrodes and
including the trap door influence will then be calculated with help of FEM simulations.
FEM simulations of characteristic parameters
The most important information that the ion trap
provides in the here interesting context is the mass
spectrum. This can be obtained from the induced
signal on the outer electrode per FFT. This in turn
requires stable harmonic oscillations. In the trap principles discussed in this paper, the for the mass spectra
important oscillations are those along the Z-axis,
which can be shown to be simple functions of the
ions’ mass to charge ratio. In order to obtain such
an oscillation, the electric potential in Z direction
has to have a quadratic dependence on the Z coordinate and a complete independence from the other
coordinates. An indication that FEM simulations
are a suitable tool for the trap simulation is therefore
the shape of the calculated electric field. We have calculated the electric field along different lines parallel
to the Z-axis through the trap for Z coordinates from
11 mm to 11 mm located at the Y-coordinates from
0 mm to 11 mm in steps of 1 mm with X-coordinates
of 0 mm, 1 mm and 2 mm. This is basically the here
interesting cuboid-shaped range in which the ions
should travel. A linear function fit over all the lines
resulted in very good linear fits with standard errors
7
for the slope of less than 0.1%, and maximum residuals of below 0.2% of the maximum field value. The
variations for the slope values when comparing all the
lines with each other where similarly below 0.2%. The
numerically calculated field is therefore to a very large
degree linear in Z direction and independent of the X
and Y coordinates, as required and described by the
field equations.
The next step is then to simulate the ion motion.
We have simulated the motion for 3 ms and calculated
the induced signal for some masses, and compared the
oscillation frequencies of the induced signal with
experimental values which were obtained with an
identical set up, including those of Koester.7
Figure 2 shows the obtained frequencies together
with a fit based on the theoretic dependency of frequency on ion mass. For the fit, only the theoretic
values were used in order to verify that they reproduce the experimental values.
The maximum deviation for the experimental
values from the fit is below 0.2% (value at m/z 118
Dalton); normally it is expected that when the mass
difference is larger with respect to the range of values
used for the fit, the error becomes larger, but here also
the value at 922 Dalton is well represented by the fit.
The next figure, Figure 3, shows an example of the
spectrum typically obtained when calculating the
FFT of the induced simulated signal (simulation
time 3 ms). The spectrum does not only show the fundamental frequency of the ion motion, but also odd
overtones. The other broad peaks with very low
intensity are the result of the ion cloud motion perpendicular to the Z-direction. Initially, the ion cloud
is quite small and moves in an oscillatory motion in
both, Y- and Z-direction. The different frequencies of
the Y- and Z-motion plus the Y-motion’s stronger
Figure 2. Calculated and experimentally determined frequencies
for ions of different mass. The fit based on the simulated values
shows how well the calculated values correlate with the experimental values (resolving power larger than 50,000 correspondingly the error is too small to be indicated in the graph), based
on the theoretic dependence of frequency on ion mass. The
values marked with “x” were taken from ref. 7.
8
Figure 3. Calculated spectrum of rhodamin B based on the
simulated ion motion, simulation time 3 ms. The odd overtones
are visible, other peaks are caused by the oscillation in
Y-direction which after these short times still contributes to
the induced signal.
dependency on the initial conditions lead to a faster
de-phasing of the ions in Y-direction. After some
time, the cloud is spread evenly along the Y-axis
with a random distribution of the velocity vectors in
that direction while being relatively concentrated in
Z-direction. Then the induced signals of the single
ions cancel each other for the Y-motion, so that the
obtained signal is only showing the Z-motion. Here,
the simulation time was due to time and memory
restrictions only 3 ms; in experiments, the measurement time is much longer (hundreds of milliseconds
up to a few seconds) in order to reduce the FFT signal
width, and then the Y-motion is not visible in the
spectra anymore so that only the fundamental frequency and the odd overtones remain. In the experimental spectra, the intensity of the second peak at
three times the fundamental frequency can be as
low as 8% of the fundamental signal’s intensity,7 in
the simulations it is larger due to the shorter simulation time (ca. 20%, see dashed line in the figure).
Figure 4 shows for comparison an experimental spectrum of reserpine. The overtones are present, but no
other signals.
The basic parameters such as ion frequency and
correspondingly ion mass can be obtained by simulation. The error when comparing with experimental
values is below 0.5%. However, the peak width and
therefore the resolving power cannot be compared.
For this, simulation times comparable with the experimental measurement times are necessary, which is
not feasible for us at the moment. Nevertheless, for
set up optimizations or testing of different device
parameters (e.g. electric or geometric), the simulations are already helpful since they show a relatively
good level of precision within their time frames. As an
application example, we investigate in the next part a
so called high field Cassinian Trap with the goal to
increase the ion oscillation frequency which is directly
related to the resolving power.
European Journal of Mass Spectrometry 27(1)
Figure 4. Experimental spectrum of reserpine. Also here odd
overtones appear, but the measurement time is long enough so
that the other peaks do not appear; the signal at 566 kHz is only
at ca. 10% intensity of the fundamental peak.
FEM simulations of a high field Cassinian trap
The ion motion’s frequency depends on the voltage
difference between outer and inner electrode; here,
one electrode is on electric ground so that only one
voltage influences the frequency. Furthermore, the
difference between the squared values of outer and
inner radius has an influence. However, both of
these values are under the square root in the frequency formula, so that changing only one results in quite
a little change of frequency. A similar problem is
known from the high field Orbitrap.11 Therefore we
tried to calculate ion trajectories with a relatively
small change of geometry and consequently a larger
change in electrode voltage. The ion inlet, the trap
door, changes the trapping field and therefore determines the initial conditions of the ion motion, especially the initial velocity vector. Subsequently, it is
quite difficult to obtain an ion trajectory that can
oscillate in Z-direction for long times.
To get an initial overview for which conditions an
ion trajectory can be obtained that does not collide
with the inner or outer electrodes, we have calculated
the ion trajectory in a simplified manner, i.e.
with help of the equations of motion alone.
Consequently, we have solved the differential equations in X- and Y-direction and determined the maximum X-coordinate (absolute values) within the first
100 microseconds, since previous simulations showed
that this time frame is long enough to get a quite
stable ion trajectory also for much longer time
frames if the ions can survive that long in the trap
without collisions with the electrodes. The larger the
initial X-coordinate, the stronger the ion trajectory
oscillates in that direction. We have chosen for all
the differential equations an initial X-coordinate of
0.3 mm. The trap door’s radius is 0.4 mm, so with
Raupers et al.
0.3 mm we concentrated on the inner central part of
the initial ion cloud.
Figure 5 shows a diagram with the initial start
location in Y-direction and the initial velocity in
X-direction resp. in Y-direction; the trap voltage is
4000 V. The colors indicate conditions for which
the maximum absolute amplitude in X-direction was
smaller than 1 mm (white area). These ions consequently do not collide with the inner electrodes,
since their smallest distance in the center of the trap
is ca. 2 mm. In Y- and Z-direction, the ions do not
exceed the initial (absolute) values (if proper initial
velocities are chosen), so that these are normally not
problematic after successful injection. Allowing a
maximum absolute X-coordinate of 1 mm is very conservative but should help to get initial conditions with
good tolerance, especially since the differential equations just calculate motion in the pure Cassinian
Field, and exclude e.g. the distortions caused by the
trap door.
For the standard trap, the ions travelling through
the trap door reach the outer electrode at ca. 13.6 mm
in Y-direction. As can be seen in Figure 5, only for
very low initial velocities this is close to the area
where the maximum X-coordinate reaches values up
to 3 mm. Such X-coordinates lead to collisions with
the inner electrodes if they appear at certain
Y/Z-coordinates, especially in-between the central
Figure 5. Diagrams showing the maximum X-coordinate
(absolute values) within the first 100 us of the ion motion calculated from the differential equations. The white-grey area
shows conditions for which the maximum reached 1 mm or less,
the next light grey area with diagonal fill pattern (lower left to
upper right) shows the same for a maximum X-coordinate
between 1 mm and 2 mm, followed by the area showing a
maximum X-coordinate between 2 mm and 3 mm; the dark grey
area shows values larger than 3 mm. The initial conditions were
0.3 mm for the X coordinate, 0 m/s for the initial velocity (upper
graph: in X-direction, lower graph: in Y-direction), and the
conditions given on the axes. A maximum absolute X-coordinate
of less than 2 mm means that the ions do not collide with the
inner electrodes since the minimum distance between the inner
electrodes is 2 mm.
9
electrodes near the center of the trap; towards the
trap’s ends, the distance in-between the inner electrodes is larger (see Figure 1). However, in the simulations we encountered no such collisions in the
first 3 ms of simulation time so the maximum
X-coordinates were only reached at un-critical locations. This shows that the trajectories based on the
differential equations alone are only of limited help.
However, since they can be calculated much faster
than the trajectories considering the complete trap
geometry, as it is the case in the FEM simulations,
we still used them to get a quick overview of suitable
initial conditions. Examples regarding how these stability diagrams change with trap dimensions and also
initial conditions can been found in Gunzer.14
Figure 6 shows the corresponding diagrams for a
trap with a voltage of 8000 V applied to the inner
electrodes, and slightly smaller outer electrode distances from the trap’s center line achieved by setting the
outer radius ao to a value of 11 mm (instead of 12 mm
in the original trap) and the stretching factor B to
1.975 (otherwise the trap gets stretched very strongly
into y-direction for low absolute z-values.). Since it is
the difference of the squares of outer and inner radius
which is influencing the oscillation frequency, smaller
changes of the outer radius can achieve larger changes
of this frequency in comparison with changing the
inner radius. This trap should increase the frequency
for the simulated rhodamin B ions by 60.75%. The
trap door location is the same; the ions enter the then
smaller trap at a y-coordinate of 11.6 mm, i.e., at a
lower Y-distance due to the reduced outer electrode
distance from the trap center line. In this case, there is
a quite large range of initial velocities in X- and
Y-direction for which the maximum absolute
X-coordinate remains below 1 mm in Figure 6. In
Figure 6. Same as Figure 5, but for a trap with trap voltage
-8000 V, ao of 11 mm and B of 1.975. The ions reach the trap
at Y ¼ 11.6 mm, so for low initial velocities the trajectories
have always a maximum X-coordinate much smaller than the
minimum X-coordinate of the inner electrodes (white-grey
shaded area without fill pattern).
10
order to have a trap door that does not disturb the
Cassinian Field too much, we have chosen a trap
door voltage of only -240 V. If the ions start from
an electric potential of only -80 V (original trap:
-150 V), they reach a maximum velocity of ca.
8000 m/s inside the trap door, and then get decelerated towards its ends.
Here, another difference in comparison with a high
field Orbitrap becomes clear. In the high field
Orbitrap, a stronger acceleration of the ions is necessary to force them on an orbital motion if the magnitude of the inner electrode voltage is increased. In
the Cassinian Trap, the for the mass analysis important motion can be achieved even with lower injection
voltages as shown in the case here, with a reduced
disturbance of the trapping field as a consequence.
The higher injection voltage in the Orbitrap also
puts tighter restrictions on the quality of the
vacuum inside the trap. The high field Orbitrap furthermore has a thicker inner electrode compared to
the standard Orbitrap. In the case investigated here,
the inner electrodes are nearly unchanged, and only
the outer electrode’s geometry is changed. Therefore,
the induced image current is not reduced due to a
closer distance to the inner electrode,11 but rather
increased due to a reduced distance to the outer electrode. Finally, as indicated before, the process of
achieving low pressure values inside the Cassinian
Trap is much less affected since there are four openings into the trap volume, compared to two in the
Orbitrap; these openings also have here a reduced
area in the high field version, albeit only slightly
reduced. With too few or too small openings, it
becomes increasingly difficult to reach the low pressure values required in such traps so that e.g. strong
"baking out" at high temperatures for long times
becomes necessary.11 Another interesting difference
is that in the Cassinian Trap, the increase of trapping
voltage leads to relaxed initial conditions necessary
for successful trapping, as can be seen when comparing Figures 5 and 6.
Although Figures 5 and 6 seems to allow a quite
large parameter range to achieve stable trajectories, it
is still very difficult to achieve such trajectories in the
FEM simulations. One key parameter was the voltage
difference crossed inside the trap door tunnel. This
difference had to be reduced for successful trapping
of ions. The resulting influence of the trapdoor on the
ions’ velocity is then at the trapdoor’s end quite small,
and the main factor that remains is the Cassinian
Field which reaches to a certain extent also into the
trap door. The FEM simulated ion trajectories for
such a set up did not show any collisions with the
ion trap housing or electrodes, which is consistent
with the estimations based on the stability diagram
shown in Figure 6. It shows that the initial X-velocity
is the most critical factor and has to be kept low at the
entrance to the trap. This is complicated by the fact
that the opening of the trap door leads to the
European Journal of Mass Spectrometry 27(1)
formation of an electrostatic lens, which introduces
a focal point for the ion trajectories after leaving
the trap door and correspondingly introduces a velocity component into X-direction. With the chosen trap
door voltages this influence was, however, not critical. Figure 7 shows the obtained velocities extracted
from the FEM simulations for the initial ion trajectory into the trap for the particle with the largest initial X-coordinate. This particle’s X-velocity felt the
strongest change due to the electric trap field reaching
into the trapdoor so that the velocities are affected
already at a Y-coordinate of 12 mm. As can be seen,
near the entrance into the trap (Y-coordinate of
11.6 mm) the X-velocity is still low, especially lower
than the limit required for stable trajectories (see
Figure 6) with a quite large tolerance regarding how
deep the particle has to travel into the trap before the
X-velocity becomes too high. The simulated trajectories reached minimum distances to the inner electrodes of 1.2 mm; the maximum X-coordinate was
1.8 mm but was only reached for Y-coordinates
larger than 10 mm and correspondingly far away
from the inner electrodes.
The resulting trajectory stability obtained for the
full trap set up values fit quite well to the calculations
shown in Figure 6, especially when looking at the
Y-velocity during injection which remains in the simulated trajectory quite constant at the initial value of
8000 m/s (Y-axis pointing away from the trap
center, therefore negative velocity values) and
decreases slightly to 8800 m/s near the entrance.
But also here, a change of velocity is observed
Figure 7. Graph showing the FEM-simulated X-/Y-velocity of a
particle (with initial X-coordinate of 0.3 mm in the trap door;
this is the smallest value considering all simulated particles and
is therefore affected strongest by the trapdoor and the trap field
reaching into the trap door, leading to a positive X-velocity) near
the entrance into the trap (vertical line at Y-coordinate of
11.6 mm). The important X-velocity remains low around the
entrance so that the resulting ion trajectory does not collide with
the inner electrodes (see also Figure 6). However, both X- and
Y-velocities are affected already in the trap door (Y-coordinate
larger than 11.6 mm), which is the reason for the fact that solving
the equations of motion in an ideal field is helpful but not
sufficient for a realistic set up.
Raupers et al.
before the ions enter the trap volume which means, as
described before, that the information obtained by
the equations of motions is limited; however, it was
nevertheless helpful to obtain suitable start conditions
for the FEM simulation.
Figure 8 shows the calculated mass spectrum
obtained with the induced signal in the FEM set up.
The frequency increase is consistent with the theory;
the fundamental peak shifted from 200 kHz to
321 kHz which is an increase of 60.5%. The quality
of the spectrum is comparable to that of Figure 3 with
larger overtone intensities due to the shorter simulation time (only 1 ms). An interesting fact is that the
induced signal initially decays in amplitude, but then
reaches a relatively constant amplitude after ca.
0.2 ms, around which it slightly oscillates. That
means that the de-phasing of the ion cloud in Z-direction seems not to influence the induced signal anymore after a relatively short time, which is beneficial
since a decaying signal amplitude contributes to the
FFT signals width (see also Raupers et al.10) In facts,
for the high field set up with the changed trap door
voltages the space covered by the trajectories was
even more compact than that of the simulations
using the experimental set up’s parameters.
This proof-of-concept shows that also for the
Cassinian Trap it is possible to develop a high field
version with increased fundamental frequencies for
the analytic ion oscillation, which directly influences
the resolving power. Here we chose to increase the
voltage much stronger so that the geometry changes
only little, but other variations are possible.
Interesting is that for -8000 V it seems that this geometry (outer radius 11 mm, B factor 1.975) is the best
that can be reached; for example, a further reduction
Figure 8. Calculated spectrum of rhodamin B based on the
simulated ion motion for the high field trap. The frequency
increase is consistent with the theory (increase of 60.5%).
The inset shows the projection of the ion trajectories on the
X-Y-plane in order to show their stability since at all simulated
times they are relatively far away from the electrodes, especially
regarding the smallest distance between the inner electrodes
reached at z ¼ 0 which is shown here.
11
of the outer radius then leads to simulated ion trajectories which collide with the inner electrode after less
than 50 microseconds even if the stability diagrams
based on the differential equations without considering the trap door might indicate a different behavior.
The disturbance by the trap door is too strong, especially the initial X-velocity increases quickly if the ion
entrance point into the trap is moved further down to
lower Y-coordinates which in turn is achieved by
moving the trap door to larger Z-coordinates. Then
the information obtained by calculating a motion in
the ideal Cassinian Field cannot be carried over to the
more realistic FEM simulation. However, for other
geometries/voltages, an even stronger increase of the
fundamental frequency by more than 60.5% might be
possible. A re-design of a trap door to a more complex set up, possibly including electrostatic lenses to
balance the field disturbance inside the trap near the
trap door exit, could proof to be helpful.
Conclusions
In this contribution we have analyzed ion trajectories
of a second order Cassinian ion trap with trap door
inlet using FEM simulations as well as numerical solutions of the equations of motion. It was shown that
the FEM simulations can reproduce basic quantities
such as field independence regarding certain directions and frequency of ion oscillations quite well,
the errors in comparison with experimental values
are typically below 0.5%. Consequently, FEM simulations have then been used in order to test the concept of a high field Cassinian ion trap with increased
resolving power by increasing the ions’ oscillation frequency. The numerical solutions of the equations of
motion without considering the influence of the trap
door can give a good overview regarding suitable initial conditions in a much faster way than the more
complete FEM simulations, especially when the trap
door voltage is relatively low. An increase of the fundamental ion frequency for rhodamin B (mass 479.08
dalton) by 60.5% could be simulated for a trap with
-8000 V on the inner electrodes and relatively small
changes of the trap geometry, leading to stable ion
trajectories that are compact enough so that the
ions do not collide with the inner electrodes during
the simulation time (1 ms in steps of 1 ns). Different
trap set ups with stronger geometry changes and less
voltage supplied should be tested in order to see if an
even larger increase of the fundamental frequency is
possible. The simulations showed that the critical
factor is the initial disturbance of the ion trajectory
when leaving the trap door, since the equations of
motion based on the ideal Cassinian Field typically
show a relatively large tolerance towards initial conditions, leading to stable ion trajectories when
neglecting this disturbance. Especially the width of
the trapdoor in relation to its Y-location allows
ions not travelling in the middle of the trap door at
12
(X,Y,Z) ¼ (0,Y,Z) coordinates to reach too high
velocities in X-direction when leaving the trap door,
so that they cannot be trapped anymore for long
times.
The Cassinian ion trap of second order at least in
principle allows for ions starting their motion within
the trap. This is not possible with the current set up;
an ionisation method like e.g. laser radiation would
have to be brought into the trap through e.g. an extra
hole in the outer electrode. In theory this is possible
although currently maybe not practical, and then a
much stronger increase of resolving power should be
achievable.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of
this article.
Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
ORCID iDs
Juergen Grotemeyer
https://orcid.org/0000-0002-89914962
Frank Gunzer
https://orcid.org/0000-0002-4310-4948
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