Introduction

FT- ICR mass spectrometry (FT-ICR MS) offers the highest mass accuracy and resolution of any mass spectrometer [1]. These features and others have made FT-ICR MS an important tool in many areas of study, including proteomics and petroleomics [2,3,4,5,6]. Petroleum samples in particular require high mass accuracy, resolution, and dynamic range because of the high peak density in their mass spectra. As ICR technology has improved, there has been increasing interest in designing new analyzer cells to obtain the highest resolution and the best mass accuracy.

Ions contained in a strong magnetic field will execute circular orbits, known as cyclotron motion, in a plane perpendicular to the field, with a frequency given by:

$$ {\omega}_c=\frac{\mathrm{qB}}{\mathrm{m}} $$
(1)

where q is the charge on the ion, B is the magnetic field acting on the ion, and m is the ion’s mass. To incorporate this principle in a mass spectrometer, the ions must be trapped along the magnetic field axis during the measurement of their cyclotron frequency, requiring the application of a static trap** field. In the quadratic trap** potential approximation, the trap** field perturbs the ion’s motion, adding another periodic mode, called drift or magnetron motion, with a frequency given by [7]:

$$ {\omega}_{-}=\frac{E_r}{rB} $$
(2)

where E r is the radial component of the trap** electric field that repels ions away from the principal axis of the cell, and r is the radial displacement from that central axis. The frequency of the signal that is measured in a conventional FT-ICR MS is approximately equal to the cyclotron frequency (ωc) minus the magnetron frequency (ω-). This is known as the observed frequency or the reduced cyclotron frequency (ω+). For cells with an ideal quadratic trap** potential for which the ratio of electric field radial component to the radial displacement, \( \frac{E_r}{r} \), is a constant over the volume of the cell, the spatial distribution of ions (generally, an ellipsoid of revolution) is maintained during the detection period. This provides a transient with a long period of duration, from the differential charge induced on an opposing set of detection electrodes. For simple analyzer cells (either cubic or cylindrical), additional, anharmonic terms are added to the gradient in the trap** field, meaning \( \frac{E_r}{r} \) (and thus ω-) is not constant throughout the analyzer cell. Consequently, the ion packet loses spatial coherence as ions oscillate along the trap** axis, due to the inhomogeneous nature of the radial electric field as a function of displacement along the z-coordinate [1]. This leads to the formation of a comet structure that accelerates dam** of the transient signal, as the resulting symmetric distribution of ions does not produce an induced charge signal at ω+ with dipolar detection circuits [8].

Analyzer cells with hyperbolically shaped electrodes produce a purely quadratic trap** potential, and have been employed by physicists who make high precision mass measurements of atomic species [9]. However, the trap** volume of such a cell is small, and is not useful for analytical applications. To improve the resolution and mass accuracy of FT-ICR, efforts to improve analyzer cell design have focused on shimming the electric field to make it more closely match the ideal quadratic trap** potential, through static or dynamic harmonization [9,10,11,12,13]. This reduces variation in the magnetron frequency, thus improving ion cloud coherence. A more coherent ion cloud can be measured for a longer period of time, increasing resolution.

In stark contrast to this design philosophy, a recent publication described an analyzer cell that deviates from the quadratic trap** potential. The narrow aperture detection electrodes (NADEL) cell [14], or NADEL, utilizes flat detection electrodes that extend in from the cylindrical excitation electrodes toward the center of the analyzer cell [15]. These electrodes create a significant perturbation to the quadratic trap** potential, Figure 1, and the electric field gradient, \( \frac{E_r}{r} \) , varies considerably over the volume of the cell. Using both dipolar and quadrupolar ion detection, NADEL cells were able to demonstrate long transients and high resolution, as well as a capability to detect the near unperturbed cyclotron frequency, ω+ + ω- = ωc [15]. The latter feature makes the mass calibration less dependent on trap** voltage and space charge, and thus improves the frequency measurement precision and, consequently, the mass accuracy for experiments with strong variation of ion population.

Figure 1
figure 1

Comparison of electric fields in SIMION models of a 3D quadratic trap** potential, NADEL cell and cubic cell. The 3D quadratic trap** potential causes no (a) angular or (b) axial variation in Er/r, and there are no perturbations to in the (c) potential contour lines. In fact, all data points are the same because Er/r is a constant for this cell. The NADEL cell, in contrast, has simple trap** electrodes and inward protruding detection electrodes that cause (d) axial and (e) angular variation in Er/r, respectively. The (f) potential contour lines at the center of the analyzer for the NADEL cell show the perturbations caused by the protruding detection electrodes. A cubic cell produces (g), (h) variations in Er/r and (i) perturbations to the contour lines that are very similar to those produced by the NADEL cell

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This is not the first demonstration of the generation of the unperturbed cyclotron frequency, or even observing it as the primary signal. This concept was introduced by Schweikhard et al. in 1989 when they showed theoretically and experimentally that by using quadrupolar detection, one could generate two signals corresponding to ω+ – ω- and ω+ + ω-, the latter being equal to the unperturbed cyclotron frequency [16]. It was later shown that by using two pairs of electrodes, the signal for the unperturbed cyclotron frequency could be increased, while the ω+ – ω- signal was decreased [17]. If the magnetron radius is equal to the cyclotron radius, the ω+ – ω- signal would decrease to zero [17]. However, these experiments used an analyzer cell with hyperbolic-shaped electrodes and were performed only on low mass ions. Moreover, the intensity of second harmonic, 2ω+, always remained higher than 20%–30% w.r.t. ω+ + ω- signal, which is undesirable for real-life biological applications. Since that finding was published, this method has not been pursued in conventional ICR cells.

The NADEL cell design with two pairs of detection electrodes introduces significant anharmonic terms to the trap** field. Compared with other unshimmed analyzer cells, for example the cubic cell, the cyclotron frequency can be observed without loss of signal or resolution. By conventional understanding and considering a tightly bunched ion distribution, one would expect the ion cloud to quickly lose coherence, leading to a swiftly decaying transient [8]. Instead, the signal was shown to be stable and produce the unperturbed cyclotron frequency, due to a unique cloud structure [15, 18]. SIMION simulations of up to 500 ions in NADEL cells suggested that phase-coherent spatial dispersion of ions in a non-linear, azimuthally dependent electric trap** fields is responsible for the observation of frequency spectra at the unperturbed cyclotron frequency [19]. Each ion in such a field is characterized by its own cyclotron frequency with a center of the ion orbit drifting about the off-axis electric center of the trap** field. As a result, most, if not all, ions in a cell will contribute to the intensity of the cyclotron frequency peak but will introduce a destructive interference for the reduced cyclotron frequency peak [18, 19]. To gain further insight into the manner in which the NADEL cell is able to achieve this performance, one needs to consider the collective motion of the ions in more detail, using multi-particle computer simulations with more features than offered by SIMION. Such simulations have been shown to be a useful approach for examining the performance of FT-ICR analyzer cells, for example to accurately model the influence of subtle effects such as space-charge and ion-induced charge interactions [20,21,22,23,24]. Simulations provide the ability to test the effects of parameters that are difficult to control experimentally, such as the precise number, initial energy, and spatial distribution of ions in an analyzer cell [25]. The particle-in-cell (PIC) method, borrowed from the plasma physics community, offers a computationally tractable way to examine motion of realistic numbers of ions while maintaining an accurate account of space charge [26,27,28]. Additionally, the PIC method has been extended to include image charge forces on surfaces of arbitrary geometry, which is crucial to understanding the signal generated on the narrow detection electrodes [29].

Experimental

Multi-particle simulations were performed on z-cluster, a Linux cluster at the University of Georgia, on nodes equipped with Intel Xeon processors, and NVIDIA K40m GPU cards. The PIC software is highly parallelized and uses the GPU cards to decrease calculation times. It also includes a parallelizable charge colocation based method for determining image charge, as described previously [29]. Simulations utilized a trap** potential defined by a SIMION model with the trap** plates set to 10 V. The simulations utilized a homogeneous magnetic field with a strength of 7 or 10 Tesla. The analyzer cell geometry is in accordance with a NADEL cell design with a diameter of 5.6 cm, a cubic cell of diameter 5.8 cm, or a dynamically harmonized cell composed of eight hourglass/leaf electrode pairs with a diameter of 6 cm. The narrow detection electrodes stick into the NADEL 1.1 cm leading to an inscribed radius of 1.7 cm. For most simulations, a monoisotopic population of 1000 ions with m/z = 524 was generated with an ellipsoidal distribution (major axis = 2.5 mm; minor axis = 0.2 mm) with the major axis parallel to the magnetic field in the center of ICR cell and given an initial velocity (energy) in a line distribution of 0.2–0.4 mm/μs (0.11–0.43 eV) in the z-axis. These parameters were chosen to generate maximum axial displacement in the NADEL cell, approximately 1.6 cm, just inside the annular trap** electrodes. Coalescence simulations instead used a 23+ charge state of cytochrome c (22 most abundant masses in a theoretical distribution) with the same initial space and energy distribution and 50 V trap** potential. This high trap** voltage accelerated the process of coalescence, and reduced the computer simulation time to a tractable range.

A broadband chirp excitation from m/z 200 to 10,000 over 5 ms with Vp-p of 25 V was used to excite ions to a higher cyclotron radius. The magnetron radius was then excited by applying DC potential of ±5 V to opposing excite plates for a duration of 500 μs after cyclotron excitation. A simulated time domain transient was collected for each experiment by summing the induced signal on opposing electrodes and subtracting the signal from orthogonal electrodes. This signal was sampled at 2 MHz for a duration of 0.5 s, followed by two zero-fills. No apodization was applied. A frequency domain spectrum was derived using FOM-AMOLF’s AWE software [30]. Unless otherwise noted, experiments were performed with a quadrupolar detection scheme. The ion cloud and the resulting image charge distribution was visualized with in-house software developed at FOM-AMOLF [21]. A Fourier transform ion cyclotron resonance mass spectrometer (SolariX, Bruker Daltonics, Bremen, Germany) equipped with a 9.4 T superconducting magnet was employed to acquire all experimental data. The FT-ICR MS was equipped with the Paracell, also referred to as DHC cell, or the NADEL ICR cell (Spectroswiss, Lausanne, Switzerland) based on a design of an open-ended cylindrical ICR cell (Ultra Cell, Thermo Scientific) as described elsewhere [7]. The NADEL ICR cell geometry represented a cylindrical surface with the four 90° wide excitation electrodes and four narrow aperture detection electrodes [15]. NADEL ICR cell had an aspect ratio of 2.5 (length of 140 mm and an inner diameter of 56 mm). The instrument was operated via standard in-built data acquisition station and instrument control software (Bruker Daltonics).

Results and Discussion

NADEL Cell with Quadrupolar Detection

As shown by Schweikhard et al. [17], the signal for the unperturbed cyclotron frequency is maximized when the center of an ion’s cyclotron orbit is offset from the center of the cell by a distance equal to the radius of its cyclotron orbit. This is consistent with experimental and computational (using SIMION) results with the NADEL cell [14, 15, 18, 19], and these conditions were examined by PIC simulations. Figure 2a shows a calculated transient from a PIC simulation of quadrupolar detection in the NADEL ICR cell, with initial conditions selected that make the cyclotron radius approximately equal to the magnetron radius. The first 0.1 s of the transient shows the expected cyclotron frequency (~293055.7 Hz) superimposed on the magnetron frequency (~25 Hz). During this time, the ion cloud retains its initial structural qualities: tightly packed ellipsoid with all ions moving in coherent motion (Figure 2c). This result is consistent with the experiments performed by Schweikhard et al. [16, 17], generating a peak at the unperturbed cyclotron frequency (ωc) and two times the reduced cyclotron frequency (2ω+), the first due to combination frequencies (ω+ + ω-), the second due to measuring the coherent ion motion twice per cyclotron orbit.

Figure 2
figure 2

Comparison of quadrupolar detection in (a)–(d) NADEL cell, (e)–(h) DHC, and (i)–(l) cubic cell. (a), (e), (i) Simulated transients and (b), (f), (j) resulting frequency spectra. Visualization of the ion positions at (c), (g), (k) t = 0.0 s and (d), (h), (l)) t = 0.5 s. The red circles describe cyclotron orbits for one or two ions in the simulation; blue circles describe the ions’ magnetron orbit. For the NADEL simulation, colored dots show ion positions over the next cyclotron orbit starting at blue and going to red

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After 0.1 s, however, the structure of the transient changes. Instead of the amplitude decay expected with dipolar detection, caused by comet formation, we see a stable signal that no longer oscillates at the frequency of the magnetron motion. By tracking the motion of an individual ion as shown in Figure 2d, we can see that ions continue to undergo cyclotron motion, but with a distribution of their magnetron phase, leading to the formation of a rotating slab structure, in line with the previously obtained SIMION results [15, 19]. The results of PIC simulations reported here confirm slab and star-like ion cloud structure formation as observed with SIMION simulations. Increased ion numbers in PIC simulations make slab formation and observation even more compelling. A video of the rotating motion of this novel ion cloud structure can be found in the Supplementary Material. During the first 0.2 s of the experiment, the ion cloud reorganizes to develop the slab distribution. Normally, the ions would spread along their cyclotron orbit, forming a comet, then a cylinder of charge once the head is reincorporated into the tail. However, because the magnetron radius is equal to the cyclotron radius, the dispersion of the ions along their cyclotron orbit does not form a cylinder. Instead, the collective motion of ions at different points in their cyclotron orbit creates the slab-like structure rotating on its major axis at half of the unperturbed cyclotron frequency. Figure 3 shows how the slab position corresponds to points in the sinusoidal image current. One can see from a comparison of the slab positions in inset A versus C that as an ion rotates 180° in its cyclotron orbit, the slab to which it belongs rotates 90°. Thus, the slab rotates at half the cyclotron frequency, (w+ + w-)/2 = wc/2. Since quadrupolar detection produces two maxima in the induced signal per slab rotation, all the ions in the slab generate a signal at the unperturbed cyclotron frequency. An additional explanation why the observed ion cloud structure rotates at half the cyclotron frequency has been published in relation to SIMION simulations of ion motion in NADEL cell [19].

Figure 3
figure 3

Simulated transient signal with corresponding ion positions showing detection of the slab-like clouds. The detection electrodes are shown as red (negative) and blue (positive) points and the transient at each time step is calculated by the image charges on the blue electrodes minus the image charges on the red electrodes. The red circle describes a single ion’s cyclotron orbit over this 4 us period. The slab rotates 180° in the same amount of time it takes an ion to complete its cyclotron orbit, resulting in the slab rotating at half of the cyclotron frequency

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This slab distribution forms when the cyclotron radius is equal to the magnetron radius, which is also the ideal ratio for generating the unperturbed cyclotron frequency through combination frequencies [17]. In the case of the NADEL cell with quadrupolar detection, the signal switches between these two modes of observing the unperturbed cyclotron frequency. Initially, ions are moving with coherent cyclotron motion at the reduced cyclotron frequency, generating a peak at ω+ + ω- and 2ω+. Within 0.2 s, however, the slab ion distribution has formed, and generates an induced current signal at the unperturbed cyclotron frequency. Interestingly, since the slab-like structure is formed when ions are spread over their magnetron orbit, the ions do not have spatially synchronized cyclotron orbits and, as a result, signal at 2ω+ is slightly less than 5% of the base peak. Figure 2b shows the peak shapes and intensities in frequency space. Because of the non-linear trap** field, magnetron frequency appears as a broad distribution, which causes the reduced frequency, ω+ = ωc–ω-, to exhibit a similarly broad shape, as can be seen in the second harmonic peak, 2ω+. In contrast, the signal at ωc has no dependence on ω-, and appears as a sharp, well-defined peak. Comparison of the signal intensities before and after the formation of the slab-like structure (Supplementary Material S1) shows that the intensity of the signal at ωc is the same for the first 0.1 s and the last 0.1 s. The signal for 2ω+, however, reduces 20-fold to only 1% of the base peak intensity. By processing the part of the transient acquired after this structure has formed, the unperturbed cyclotron frequency is the only signal of significant intensity in the spectrum, which confirms the reported earlier SIMION simulations and the experimental observations [15, 18].

Quadrupolar Detection in DHC and Cubic Cell Analyzers

The detection of the unperturbed cyclotron frequency can be achieved with other cell geometries, but we expect those with ideal quadratic trap** potentials to maintain the initial ion cloud spatial distribution, and therefore produce other peaks, e.g., at nω+ and nω+ ± kω-, in addition to the signal at ωc. This was examined with PIC simulations of both dynamically harmonized and cubic analyzer cells. Quadrupolar detection in a dynamically harmonized cell (DHC) resembles the experiments performed in a Penning trap by Schweikhard et al. [16, 17], as it has a nearly perfect 3D hyperbolic trap** field upon angular averaging [11]. Figure 2e and f show a simulated transient and the resulting frequency spectrum for quadrupolar detection in a DHC. To optimize quadrupolar detection for ωc the cyclotron radius and magnetron radius are approximately equal. The frequency spectrum shows signals at ωc, ω+, and 2ω+. The transient features are consistent for the entire simulation (corresponding to a transient duration of 0.5 s) and visualization of the ion cloud (Figure 2g and h) shows that it does not dephase during this time. The first 0.1 s of the transient generates the same frequency spectrum as the 0.4 s–0.5 s section. This demonstrates that the signal at the unperturbed cyclotron frequency is due only to combination frequency detection, as no slab-like structure is generated. Since the ions maintain coherent cyclotron orbits, they continue to generate a signal at 2ω+, increasing the complexity of the resultant spectrum. For a sample with many components, such as petroleum, this is undesirable. Although this analyzer cell does a nearly perfect job of replicating the ideal trap** field for dipolar detection, it is not ideal for use in detecting the unperturbed cyclotron frequency through quadrupolar detection since it continues to generate other frequencies.

Since the spatial dispersion of ions along their magnetron orbit is responsible for generating the unperturbed cyclotron frequency without also generating spurious frequencies, it follows that any analyzer that deviates enough from an ideal trap** field should be able to replicate the results of the NADEL cell to some extent. A cubic analyzer cell is perhaps the simplest analyzer (although implementation of quadrupolar detection would be more complicated) and should work similarly to the NADEL cell in terms of detecting the unperturbed cyclotron frequency. Figure 2i and j show a simulated transient and the resultant frequency spectrum for quadrupolar detection in a cubic analyzer cell. Like the NADEL cell transient, the cubic cell transient is clearly divided into two parts; with the cyclotron signal superimposed on the magnetron signal early in the transient, followed by a stabilized signal that does not show magnetron oscillation. Visualization of the ion cloud (Figure 2k and l) shows that the cloud disperses along its magnetron orbit, leading to the rotating slab-structure. Analysis of the signal before and after the formation of the slab shows a drastic reduction in signal from 2ω+ while ωc remains the most intense peak.

Effect of Space Charge on ωc and ω+

A major benefit of detection at ωc over ω+ is the reduced dependence on space charge. To compare the effect of space charge on these different modes of detection, simulations were performed on a distribution of seven equally spaced peptide-like isotope packets ranging in m/z from 500 to 2000. By increasing the total number of ions present in the analyzer (quadrupolar detection in the NADEL cell or dipolar detection in the DHC), the frequency change was measured. Figure 4, left shows the change in frequency for the measured signal of the ion cloud at m/z 500 in the DHC and the NADEL cell. As the number of ions increase, the frequency changes approximately 100 times more rapidly for ω+ than ωc. For ωc the frequency changes in the opposing direction compared with ω+. Figure 4, right demonstrates the change in mass for 11 peaks of poly-phosphoric acid (H3PO4)n in the mass range of 300 –1500 m/z experimentally measured with the DHC (Paracell) and the NADEL cell. As the number of ions increases, the mass (frequency) drifts approximately at least five times more rapidly for ω+ than ωc.

Figure 4
figure 4

Comparison of the effect of space charge on detection at ωc and ω+. (Left) shows results of PIC simulations. (Right) shows the mass error dependence on the number of charges (total ion current) experimentally measured using the 9.4 T SolariX FT-ICR MS with the Paracell (DHC cell, blue circles) or the NADEL cell (red circles) for the 11 monoisotopic peaks of a poly-phosphoric acid (H3PO4)n in the mass range of 300–1500 m/z in a number of single scans. The number of charges (total ion current) was varied via changing the ion accumulation time in an external ion trap before ion transfer into the corresponding ICR cells. The same set of mass calibration coefficients was employed for the experiments with Paracell (DHC) and NADEL ICR cell

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Effect of Slab Detection Mode on Coalescence

Ion cloud coalescence is an undesirable phenomenon caused by space charge interactions, which are accurately modeled by the PIC method. It occurs when ion clouds that are close in frequency phase-lock, appearing as a single peak in the mass spectrum [25]. This occurs at high ion density and results in a peak at a frequency corresponding to the average observed frequency of the coalesced ion clouds. The ions in the slab distribution have a lower charge density than in the ellipsoidal cloud typical of dipolar detection. The average radial distance between ions in the slab structure is 5.6 mm, compared with an average radial distance of 0.2 mm in the initial ion cloud before slab formation begins. Given this, one might expect slab mode detection to be less sensitive to ion cloud coalescence effects, but simulations show the onset of this behavior to occur with roughly the same number of ions in either mode of detection. To test this, we examined a simulated isotope distribution for cytochrome c (23+ charge state, m/z ~512). By increasing the number of ions in the simulation from 50,000 to 400,000 at a constant excite radius, the onset of coalescence can be observed. Figure 5 compares the mass spectra and ion positions at t = 0.05 s for increasing ion number in both the NADEL cell and a DHC. For this experiment, the goal was to compare quadrupolar detection in the NADEL cell with dipolar detection in the DHC. Both simulations are run at 50 V trap**, with a cyclotron radius equal to 20% of the cell radius (cylindrical electrodes have a 30 mm radius). The ions in the NADEL cell undergo magnetron excitation between cyclotron excite and detection, whereas the ions in the DHC are centered around the principal axis.

Figure 5
figure 5

Comparison of DHC and NADEL cell ion clouds and simulated mass spectra for cytochrome c 23+ charge state. (left) Mass spectra for DHC and NADEL cell with increasing numbers of charges present (50,000 top, 200,000 middle, and 400,000 bottom). (right) Ion positions 0.05 s after the start of detection. Colors correspond to individual m/z values as shown on the mass spectra. The DHC signal is observed at the reduced cyclotron frequency leading to higher m/z values if not calibrated

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As can be seen in Figure 5, coalescence in the DHC and NADEL cell occurs at approximately the same rate. The DHC shows partial coalescence with 200,000 charges present and full coalescence with 400,000 charges. In contrast, the deterioration of signal in the NADEL cell is more consistent with ion cloud destruction. For example, the middle panel (200,000 charges) shows that the lower abundance ions (dark blue and red dots) have lost their slab structure, but the most abundant ions (light blue and blue green dots) still retain their slab structure. For the NADEL cell, there is very little ion cloud coalescence at 400,000 charges, even though the mass spectrum shows the merging of the isotope peaks into a single peak. It is important to note that the DHC can be potentially run at 2- to 4-fold higher cyclotron radius, which should raise the charge threshold required for coalescence. A comparable simulation with excite radius equal to 40% of the cell radius is shown in the Supplementary Material (S2). The maximum cyclotron radius will be limited in the NADEL cell because of the inward protruding detection electrodes. The inscribed radius of the detection electrodes could be increased to alleviate this, but might lead to a reduction in image charge intensity and other effects that would require further study to fully understand. Conversely, the DHC has a set aspect ratio of 2.6, but the NADEL cell used in these experiments had an aspect ratio of 1.5. Extending the NADEL cell would decrease the charge density without altering its other capabilities.

With the NADEL cell, space charge can upset the balance of magnetron radius and cyclotron radius even before detection begins. For dipolar detection, this would not noticeably affect detection. However, for quadrupolar detection, this imbalance disturbs the formation of slab structures for many isotopes, leading to cloud destruction and, eventually, some coalescence. Effectively, at high charge density, the ion clouds in a NADEL cell are not participating in the same detection mode. The cyclotron frequency is still generated for high intensity and high mass ions, but the lower intensity and lower mass ion clouds do not form the necessary cloud structures for this mode of detection. Space charge can be a problem for slab formation and steps should be taken to reduce its effects during detection. This suggests that one of the anticipated advantages of quadrupolar detection in the NADEL cell, increasing the trap** voltage to hold more ions in the cell, must be tempered with space charge limitations in mind. Quadrupolar detection in the NADEL cell reduces coalescence, but does not completely remove concerns of space charge.

Effect of Ion z-axis Energy on Quadrupolar Detection in the NADEL Cell

Prior to formation of the slab structure, some reduced cyclotron coherence still exists, resulting in moderate signal intensity for the frequencies that depend on reduced cyclotron coherence (ω+ and 2ω+). If ions are quickly spread over the magnetron orbit, the signal at ωc will be strong, but the magnetron frequency, ω-, and therefore ω+ and 2ω, will be distributed over many frequencies resulting in minimal intensity for these undesired peaks. Ideally detection should not start until the slab structure has formed. One method to make the slab structure form more quickly is to increase the ion z-axis energy and the spread of energies present, as suggested by the experimental and simulated data [15, 18]. Figure 6 demonstrates how increasing the ion energy spread results in quicker dephasing of the cloud as well as greater spread of ω-, resulting in a lower intensity of 2ω+. At even higher z-axial energy spreads, the ions dephase so quickly that they do not present a coherent cloud during excitation of the magnetron radius. This leads to ions of the same mass-to-charge having different ratios of magnetron radius to cyclotron radius, causing the ions to form a misaligned slab structure that eventually becomes isotropic. Adding a delay prior to detection that allows the slab to form and stabilize before signal is collected results in spectra with minimal contributions from ω+ and 2ω+. The length of this delay will depend on the range of energies present within each ion cloud as well as the trap** voltage [15, 18].

Figure 6
figure 6

Increasing ion energy spread for 1000 ions in the NADEL cell where Vt = 50 V. As the initial energy increases, the ions can move farther along the z-axis, increasing the disparity between Er/r. This results in a wider array of magnetron frequencies, reducing the intensity of the peak at 2ω+

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Effect of Ratio of Magnetron to Cyclotron Radius

Unless the electric trap** field closely approximates a 3D hyperbolic electric field, an ion cloud will adopt a more distributed structure even in the absence of neutral collisions. If the magnetron radius is zero, the ions will form a hollow cylinder of charge when the head of the comet is reincorporated into the tail [8]. This charge distribution is isotropic, and therefore undetectable. If, however, the magnetron radius is greater than zero but less than the cyclotron radius, loss of coherence instead forms an elliptical cylinder of charge, as shown in Figure 7. This cylinder will still be isotropic for the purposes of dipolar detection, but will generate a sinusoidal signal for quadrupolar detection at the ions’ unperturbed cyclotron frequency. When magnetron radius equals cyclotron radius, the minor axis of the ellipsoid is zero, and the result is a rotating slab of charge. Further increases of the magnetron radius creates a mismatch between the alignments of the head and tail of the comet, leading to even more complex structures. Figure 7 shows how magnetron radius affects the shape of the ion cloud after dephasing.

Figure 7
figure 7

Ion cloud structure development as magnetron radius (Rm) approaches cyclotron radius (Rc). These ion clouds will produce no signal in dipolar detection after 0.15 s, but will still be detectable by quadrupolar detection, with Rm = Rc producing the most intense signal. Furthermore, the signal from the first three cloud structures will not deteriorate due to the electric fields. For m > Rc, the head and tail of the ion cloud do not meet up within the timeframe shown here, and the ions will continue to spread around the analyzer cell

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Advantages of Quadrupolar Detection in the NADEL Cell

Since the ions spread along their magnetron orbit to form the slab-like structure, a stable signal is produced as continued loss of coherence does not affect the shape of the cloud. Beyond stability, the signal generated from the slab-like structure has a few unique properties making it useful for accurate mass measurements. As with the signal generated through combination frequencies, the slab-like structure signal produces a peak at the unperturbed cyclotron frequency. This may allow mass calibration to be done without including a term for the magnetron frequency. Also, ωc does not depend on the trap** voltage as ω- does, meaning trap** voltages could be raised to increase the number of ions present in the analyzer cell, although space charge can still cause limitations. Additionally, since the magnetron frequency is not a single peak, but a very broad distribution, the signal does not produce a peak at 2ω+, resulting in a spectrum that is not more complex than its dipolar detection counterpart.

The NADEL cell is particularly suited for quadrupolar detection for several reasons. The trap** field already deviates from the conventional trap** field enough to allow swift dephasing into the slab structure. The detection electrodes are also well designed for use here. Because the detection electrodes are narrow, quadrupolar detection can be performed without needing to use the same plates for excite and detect. With the NADEL setup, there can easily be four plates for detect, two for excitation of cyclotron motion, and two for excitation of magnetron motion. Additionally, a simple implementation of quadrupolar or quadrature ion excitation scheme can be done with NADEL ICR cell, possibly further improving the performance of the observed phenomenon [31, 32]. Since the detection electrodes have lower capacitance compared with the conventional 90° electrodes and protrude into the analyzer cell, this setup can be utilized without loss in signal. Furthermore, the performance of the NADEL cell should be less sensitive to imperfections in its fabrication (machining tolerances, plate alignments, spacing), as ion behavior does not rely upon an adherence to the ideal quadratic trap** potential. In fact, a design with rectangular NADELs instead of the curved ones leads to further deviation from the quadratic trap** potential [14], which could lead to faster slab formation without higher trap** potential and improved excitation field in the plane along the magnetic field.

Conclusions

The NADEL cell design is a radical departure from high-performance ICR cell designs that seek to emulate the ideal 3D quadratic electric potential. Owing to its unique properties, the NADEL cell has renewed interest in the concept of quadrupolar detection for use in accurate mass measurements. Through particle-in-cell ion trajectory calculations we have gained additional insight into how this method of detection works in the NADEL cell. The results obtained here are in line with and extend those from the previously reported SIMION simulations for the similar NADEL cell geometry but fewer number of charges (ions) [19]. We have confirmed here that quadrupolar detection can achieve stable and long-lived induced signals, despite the large non-quadratic terms in the trap** potential, and yet it does not add complexity to the spectrum. This is achieved through the generation of a unique cloud structure that, once formed, does not undergo rapid decay. This results in long transients with highly accurate frequency (mass) measurements. Substantially reduced frequency shifts for increased ion numbers for NADEL cell compared with the DHC cell are suggested by PIC simulations and confirmed with the experimental measurements. Although very high charge density can result in slab cloud destruction, careful experiment design will make this a useful tool in FT-ICR mass spectrometry.