Tailoring fields of one-sheet and two-sheet planar ion trap mass analyzers

International Journal of Mass Spectrometry 446 (2019) 116218

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International Journal of Mass Spectrometry journal homepage: www.elsevier.com/locate/ijms

Tailoring fields of one-sheet and two-sheet planar ion trap mass analyzers Nishant Goyal, Atanu K. Mohanty* Department of Instrumentation and Applied Physics, Indian Institute of Science, Bengaluru, 560012, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 April 2019 Received in revised form 9 August 2019 Accepted 4 September 2019 Available online 10 September 2019

In this paper, a simulation study on tailoring the fields in planar ion trap geometries for making them suitable for mass analysis is presented. Two different planar trap geometries were considered: the first is a One-Sheet Ion Trap Geometry in which the ions are trapped off-plane and the second is a Two-Sheet Ion Trap Geometry in which the ions are trapped in between the two sheets. Both DC and RF potentials were used to trap ions. The fields were tailored to obtain linear trapping fields in these two geometries. This was done by splitting the central electrode into segments and applying suitable DC potentials to them. The potentials were computed using a least square method. The simulations were carried out considering a printed circuit board (PCB) with a Teflon base. The One-Sheet Ion Trap Geometry consists of five electrodes, of which the central electrode is segmented. In Two-Sheet Ion Trap Geometry, each sheet consists of three electrodes, of which the central electrode is segmented. The method outlined in the study is able to tailor fields to be linear as well as mildly superlinear. © 2019 Elsevier B.V. All rights reserved.

Keywords: Planar ion traps Mass analyzer Segmented electrode geometry PCB ion trap BEM with dielectrics

1. Introduction A planar ion trap consists of planar surface metallic electrodes fabricated on an insulator, an example of this being a printed circuit board (PCB) [1e4]. Two configurations have been discussed in literature: one is a single-PCB design and the other is a two-PCB design. The trapping of ions, which occurs off-plane in the case of a single-PCB design and in between the two sheets in the case of a two-PCB design, is affected by applying either RF only [5,6] or a combination of both DC and RF potentials on the electrodes [1e4,7,8]. Planar ion traps are easy to fabricate, can be scaled up or miniaturized and provide direct access to stored ions [9]. These traps have been used for a wide variety of applications such as quantum information processing [10e12], optical spectroscopy [13,14] and ion crystals study [4,15,16]. The electrode design in planar trap geometries varies from linear structures such as variants of the five-wire design [1e3,5,7,17e19] to circular [4,14,20e24] and elliptical [15,25] configurations. In addition to experimental studies, simulation studies have also

* Corresponding author. E-mail addresses: [email protected] (N. Goyal), [email protected] (A.K. Mohanty). https://doi.org/10.1016/j.ijms.2019.116218 1387-3806/© 2019 Elsevier B.V. All rights reserved.

been reported in literature to understand the behaviour of planar traps. These include studies on five-wire geometry, where the assumption of electrodes of infinite length has been used to study trapping conditions [26e32]. Another group of studies include those which consider circular ring geometries [27,28,33,34]. There have also been a few reports on the use of planar electrode ion traps for mass analysis. Austin and co-workers [35e40] have demonstrated the trapping of ions between a two-plate assembly, micro-fabricated with concentric electrode rings on each plate with the two plates facing each other. In these traps, the trapping of ions is affected by RF-only fields. Another similar two-plate geometry that has been proposed consists of electrodes that are rectangular. Such geometries have been investigated by Austin and co-workers [41e44] and Ding and co-workers [45e47]. In these geometries, the electrodes are excited by both DC and RF potentials. A single-sheet mass spectrometer has been reported in an experimental study by Pau et al. [9]. Their trap design consists of coplanar ring electrodes fabricated on a PCB. Here, the trapping is affected by the application of both DC and RF potentials to the different electrodes. The mass analysis was achieved through the mass selective instability mode [48]. As reported by Pau et al. [9], the performance of their trap was limited by nonlinearity of the trapping field. The effect of field inhomogeneities on trap performance have

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been well documented for three-dimensional (3D) geometries such as the Quadrupole Ion Trap (QIT) [49e51] and the Cylindrical Ion Trap (CIT) [52e55]. The field nonlinearities have been shown to result in shift in secular frequency [56], shift in ejection voltage [57], appearance of harmonics [58,59], coupling of radial, axial and RF frequencies [60] as well as nonlinear resonances [61]. Similar studies have also been reported for Linear Ion Trap (LIT) [62,63]. Some recent studies include [64,65]. In view of this, efforts have focussed on tailoring the field in 3D ion traps so as to make the field to be linear or mildly super linear [66e70]. The motivation of the present study is to provide a systematic method to linearize the field in planar ion traps. The proposed method is general and can be used to tailor the field to be mildly superlinear too. In this study first, an investigation on a single-sheet structure was taken up. It has electrodes similar to the five-wire design reported in Reference [5]. This trap was seen to have nonlinear fields. The studies on this trap were carried out to find out what needed to be done to linearize the field. For this reason, this geometry was referred to as Test Ion Trap Geometry. Using the insights gained from the study on the Test Ion Trap Geometry, two commonly-used planar ion trap geometries were taken up for linearizing their field. These include a single-sheet and a two-sheet design. These are referred to as the One-Sheet Ion Trap Geometry and Two-Sheet Ion Trap Geometry respectively, in this study. In these two trap geometries, both DC and RF potentials were used for trapping ions. It was seen that the limitations of the Test Ion Trap Geometry were overcome by linearizing the fields in these two geometries. The tailoring was done by the use of split electrodes on which appropriate DC potentials were applied. This technique of tailoring fields has been proposed and used in earlier studies [35,71]. As a caveat, it is emphasized that the present study is limited to describing a method to linearize fields in the two planar trap geometries. The performance evaluation of these traps has not been undertaken and is beyond the scope of this present study. The Test Ion Trap Geometry and the two planar trap geometries taken up for investigation are discussed in Section 2. Section 3 discusses the different computational methods used. Section 4 presents Results and discussions. Finally, Section 5 presents the Concluding remarks. 2. Geometries considered Three geometries were investigated in this study. These include the Test Ion Trap Geometry, the One Sheet Ion Trap Geometry and the Two-Sheet Ion Trap Geometry. As mentioned in the Introduction, the Test Ion Trap Geometry was studied first. Its limitations led to the evolution of the other two geometries. The parameters of these three geometries are discussed below. 2.1. Test Ion Trap Geometry The top and lateral views of the Test Ion Trap Geometry are shown in Fig. 1. This geometry is a modification of the five-wire geometry presented in Reference [5]. In this geometry, the outer two electrodes were grounded; the inner two electrodes were electrically shorted and an RF potential (VRF ) was applied on them. To confine the ion in the x direction, the central electrode was split into three segments. Of the three segments, the two end segments were electrically shorted and a DC potential (VDC ) was applied on them; the central segment was held at ground potential. The trap consisted of metal electrodes of negligible thickness placed on a dielectric base. In the simulations, the dielectric was

assumed to be Teflon (with the dielectric constant k ¼ 2:1). The dimensions of the dielectric board were 30 mm by 30 mm with a thickness of 2 mm. The two outer and two inner electrodes were 4 mm wide and 20 mm in length. The gap between adjacent electrodes was 1 mm. The three split electrodes were 4 mm wide, with the two end segments having a length of 2 mm and the third central segment having a length of 14 mm. The separation between these segments was 1 mm. The stored ions were assumed to oscillate at a height (h) above the trap, as indicated by a dashed line in Fig. 1b. The xz plane of ion oscillations at this height was referred to as the trapping plane. 2.2. One-sheet ion trap geometry Fig. 2 presents the top view of the One-Sheet Ion Trap Geometry. Like the Test Ion Trap Geometry, in this case too, the outer two electrodes were grounded; the inner two electrodes were electrically shorted and a VRF was applied on them. The difference between this geometry and the Test Ion Trap Geometry is that, here, the central electrode is split into 21 segments. The central segment was kept at ground potential while all the other segments had suitable DC potentials applied to them symmetrically on both sides of the central segment. The trap consisted of metal electrodes of negligible thickness placed on a dielectric base. In these simulations also, the dielectric was assumed to be Teflon (k ¼ 2:1). The dimensions of the dielectric board were 30 mm by 30 mm with a thickness of 2 mm. The outer two and the inner two electrodes were 4 mm wide and 24 mm long. The gap between adjacent electrodes was 1 mm. Each of the 21 segments were 4 mm wide and 0.8 mm long with a separation of 0.2 mm. 2.3. Two-sheet ion trap geometry The top and lateral views of the Two-Sheet Ion Trap Geometry are shown in Fig. 3. This geometry differs from the One-Sheet Ion Trap Geometry in three ways: (1) This geometry consisted of two parallel PCBs, with identical electrode designs, facing each other. (2) Each PCB consisted of only three electrodes. The two outer electrodes were energized by RF potential, similar to that used in the Linear Ion Trap (LIT) [62,63]. With reference to Fig. 3b, the electrodes ElectrodeA and ElectrodeC were electrically shorted, as were the electrodes ElectrodeB and ElectrodeD. The VRF was applied across these shorted electrode pairs as shown in Fig. 3c. The central electrode was split into 25 segments and had suitable DC potentials applied on them symmetrically on both sides of the central segment. (3) None of these segmented electrodes were at ground potential. The same DC excitation was applied to electrodes on the second sheet too. All the electrodes on the two PCBs were metal electrodes of negligible thickness placed on a dielectric base. In these simulations also, the dielectric was assumed to be Teflon (k ¼ 2:1). The dimensions of each dielectric board was 40 mm by 20 mm with a thickness of 2 mm. The RF electrodes were 5 mm wide and 32 mm long. The gap between adjacent electrodes was 1 mm. Each of the 25 segments were 5 mm wide and 1 mm long with a separation of 0.25 mm. The two PCB sheets were separated by a distance of 6 mm. 3. Computational methods In this section, the computational methods for analyzing three dimensional traps (lacking rotational or translational symmetry) has been described in detail not only for pedagogical reasons but also since such a description is not available in a single source in the mass spectrometry literature.

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Fig. 1. Top and lateral views of the Test Ion Trap Geometry. (a) Top view and (b) Lateral view. Indicated in the figures are the axes, dimensions and excitations given to the different electrodes.

be used for computing the ion trajectory in the trap. For this study, a general-purpose simulator was developed. This simulator has the ability to analyze arbitrary three-dimensional (3D) structures with arbitrary waveforms applied to the electrodes. It also takes the dielectric interfaces into account. 3.1. Boundary element method

Fig. 2. Top view of the One-Sheet Ion Trap Geometry. Indicated in the figure are the axes, dimensions and excitations given to the different electrodes.

The computation of fields is necessary to evaluate the ion trajectories in the trap. The fields are produced due to the applied potentials on the trap electrodes. In addition to DC, the fields have a time-varying RF component. Several methods to compute field exists in the literature, such as the Finite Difference Method [72], the Finite Element Method [73] and the Boundary Element Method [74]. In this study we have used the Boundary Element Method (BEM). To compute the fields, the BEM first computes the surface charge distribution on the electrodes and dielectrics due to potentials applied on the electrodes. The charge distribution is calculated separately for DC and RF potentials. In the case of the RF potential, for ease of computation, the charge distribution and resulting field are obtained for unit applied potential. This field is multiplied with the RF waveform to obtain the time varying field. This approach is referred to as the quasi-electrostatic approximation [75], which is valid when the size of the structure is much smaller than the wavelength at the applied frequency. Once the surface charge distribution is known, the potential and field anywhere in the space can be determined using superposition. This can

In the BEM, a given surface is meshed into small polygons called elements. In the current study, the surfaces were meshed into rectangular elements. The BEM computes the surface charges by imposing two different kinds of boundary conditions: one for the conductor and one for the dielectric. At the centroid of each element on the conductor surface, the sum of the potentials due to all the elements (both conductor as well as dielectric) has to be equal to the applied potential. At the centroid of each dielectric element, the ratio of the normal field outside the dielectric element to the normal field inside is related to the ratio of the dielectric constants. Both these normal fields are due to contributions from all the elements (conductor and dielectric) in the trap. In the simulations, the different surfaces (conductor and dielectric) available in space were not meshed in any particular order. Thus, the sequence of elements indexed in the whole mesh (from all surfaces) depended on the sequence of surfaces selected during meshing. Fig. 4 shows two arbitrary elements with indices i and j. These elements could be either on the conductor or on the dielectric surface. The unknown charge on element j was denoted by qj, j ¼ 1; 2;/N, where N is the total number of elements (including all the elements on the conductor and dielectric). A unit outward normal b on the ith element is required in case the ith element lies on vector n a dielectric surface. On the ith element, i ¼ 1; 2; /N, the boundary conditions for potential and normal electric fields were tested. The testing of boundary conditions for all N elements led to N simultaneous equations in N unknown charges. Once these equations were solved to determine the unknown charges, the surface charge distribution became available to compute the potential and field at any point in space. The boundary condition for the potential (on the conductor element) or for the normal field (on the dielectric element) provided one equation for each element. The forms of the equations obtained were different for the conductor and dielectric elements,

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Fig. 3. Top and lateral views of the Two-Sheet Ion Trap Geometry. (a) Top view of a sheet, (b) Lateral view and (c) Lateral view with electrical connections. Indicated in the figures are the axes, dimensions and excitations given to the different electrodes.

the charge on element j (which could either be a conductor or dielectric element) is now considered. On the element i, the potential Fij due to the charge qj on the jth element is

Fij ¼ Lij qj ;

(1)

where Lij is the potential on the ith element due to a unit charge on the jth element. Here, Lij is the potential Green's function relating the uniformly distributed charge on element j to the potential at the centroid of element i. When j ¼ i, the potential on the ith element due to a unit charge on itself will be referred to as Lii in the discussions below.

Fig. 4. The elements on two arbitrary meshed surfaces.

as discussed below. In the implementation of the simulator, separate records for each ith element was kept depending on the surface (conductor or dielectric). The total number of elements formed on all the conductor and dielectric surfaces were nC and nD respectively. The total number of elements formed on all the surfaces in the structure was N ¼ nC þ nD .

3.1.1. Equation for the conducting element The contribution of potential on the conductor element i due to

The total potential on the ith element (Vi ) is the sum of individual potential contributions due to charges on all the N elements including element i itself and is given as N X j¼1

Fij ¼

N X

Lij qj ¼ Vi :

(2)

j¼1

The potential Fij can be obtained from point charge approxiq

mation as Fij ¼ 4p1ε0 rijj , where ε0 is the permittivity of free space and rij is the distance between centroids of the elements i and j. However, an exact formula [74] can be derived considering that the charge is uniformly distributed on each element. A brief outline of the derivation is presented below. Fig. 5a shows an arbitrary conducting surface meshed into polygonal elements. Since a polygon can be described as a sum and

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due to charge qi present on the element i itself while the Fig. 6c shows the field developed on the ith element due to charge qj on another element j. When a unit charge is present on element i, the magnitude of the normal component of field on itself is given by Gauss's law as Gii ¼

Fig. 5. Calculation of potential at a point due to a uniformly charged element. (a) An ith quadrangular element on a charged surface (b) A triangle formed on the ith quadrangular element.

difference of triangles, the derivation below focusses on the potential due to a uniformly charged triangle. It was assumed that the triangles so formed have the same charge distribution s as on the polygonal element. Fig. 5b shows a point P lying at a distance h directly above one of the vertices of DAOB (as denoted by point O). It also shows several distances (ha , hb , la , and lb ) that appear in the formula below. The distances of vertices A and B from point O are la and lb respectively; from point P, the distances are ha and hb respectively. A perpendicular (of length p) was dropped from point O to point M on side AB (of length c). Following Reference [74], the potential UðPÞ at point P due to a uniform surface charge distribution s on DAOB can be expressed as



UðPÞ ¼







s ha þ hb þ c  hU ; p ln ha þ hb  c 4pε0

(3)

where U is the solid angle subtended at P by DAOB and s ¼ q= S, where q and S are the charge and area of DAOB respectively. The potential at O, that is UðOÞ, was obtained by setting h ¼ 0 in Eq. (3). From Fig. 5b, this gives

 s la þ lb þ c : UðOÞ ¼ p ln la þ lb  c 4pε0 

(4)

It should be noted that there is no singularity even at O. Having obtained a formula for the potential at P on account of a uniformly charged triangle DAOB, it is easy to obtain the potential at any point in space due to a uniformly charged polygon. This is because, as has been noted earlier, a polygon can be represented as a sum and difference of triangles. This enables the computation of Lij and Lii .

3.1.2. Equation for the dielectric element Next, the field on the dielectric element i due to a charge on the element j (which could be either a conductor or a dielectric element) is considered. On the dielectric surface, the field on the ith element (Eij ) due to a charge qj on the jth element is

  b qj ¼ Gij qj ; Eij ¼ Gij : n

(5)

where vector Gij is the field at the ith element due to a unit charge b is the on the jth element, Gij is the normal component of Gij and n unit vector in the direction of the normal from the ith element. Fig. 6 shows the normal field components at the centroid of the ith element, lying at the interface of two dielectrics with constants b from dielectric k2 to k1 k1 and k2 . The direction of the unit normal n is shown in Fig. 6a. Fig. 6b and c show contrasting situations when the charge is on the element itself or on another element. Fig. 6b the normal field

1 2ε0 Si

[76], where Si is the area of the ith element.

However, as shown in Fig. 6b, the directions of the fields are exactly opposite at the interface along the normal direction. When a unit charge is on another element j, the situation is different: the field (Gij ) is unidirectional and continuous along the interface, as shown in Fig. 6c. The field was computed as the negative gradient of potential (Lij ) by following the method outlined in the previous section (Section 3.1.1). To obtain the equation corresponding to the ith element, the boundary condition for normal field at the dielectric interface is imposed. The condition is that the ratio of the normal components of fields in the two dielectric media is the inverse ratio of their respective dielectric constants. The normal components of fields are related as

k1

N  X

Gij qj þ Gii qi



¼ k2

j¼1;jsi

N  X



Gij qj  Gii qi :

(6)

j¼1;jsi

The summation terms on both the left- and right-hand sides correspond to the contributions made by charges present on all the elements except the element under observation (element i). After rearranging the terms, N N X k1  k2 X Gij qj þ Gii qi ¼ 0 ¼ Jij qj ; k1 þ k2 j¼1;jsi j¼1

(7)

where Jij is the matrix element used for Eq. (7), and is given as

Jij ¼

8 > > > > <

1 ; 2ε0 Si

> > k  k2 > > G ; : 1 k1 þ k2 ij

j¼i (8) jsi

Having presented the equations for both the conductor and dielectric elements, it should be noted that each element (conductor or dielectric) provides one equation involving all the N unknown charges q1 , q2 , / , qN . By simultaneously solving the N equations in N unknowns, the unknown charges q1 , q2 , / , qN are obtained. 3.1.3. Computation of potential and field The obtained charge distribution is used to compute the potential and the field. As a matter of practice, the potential and field are obtained for unit potential. These are then multiplied with the actual applied potential to get the actual potential and field at any point in space. For the DC potential, it is straight forward whereas for the RF potential, it is multiplied by a time-varying waveform [75]. In practice, the fields are obtained separately for the applied VDC and VRF . In order to obtain the field due to VDC , a unit potential is applied on electrodes labelled as DC. All the other electrodes, including the ones labelled as RF, are kept at ground potential. Similarly, to obtain the field due to VRF , a unit potential is applied on the electrodes labelled as RF. All the other electrodes, including the ones labelled as DC, are kept at ground potential. The fields so obtained are multiplied with the respective DC and RF potentials to obtain the effective fields at any point in space. In the cases of the One-Sheet Ion Trap Geometry and the Two-

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Fig. 6. The normal electric field components at the dielectric element. (a) The normal to the element, (b) The field due to a charge on itself and (c) The field due to a charge on another element.

Sheet Ion Trap Geometry, the field due to VRF is computed as mentioned above. However, a slight modification is required to obtain the potential and field due to the DC potential. An array of different DC potentials Vapplied , to be applied on the each segment of central split electrode, is obtained from a least square method [77]. The field is computed due to the array Vapplied and not due to unit potential. All the other electrodes, including the ones labelled as RF, are kept at ground potential.

the ion. The pseudopotential depends on the mass-to-charge ratio (mion =Qion ) of ion, as seen from Eq. (10). The shape of the pseudopotential well and the equilibrium point of the ion changes with this ratio. The combination of VDC (or Vapplied ) and VRF gives a deeper pseudopotential well and increases the strength of trapping as compared to VRF alone. 4. Results and discussions

3.2. Trajectory of ion motion The equation of motion for an ion [75] is given as

mion

d2 r ¼ Qion Eðx; y; z; tÞ dt 2

(9)

where mion is the mass of the ion, r is the position vector of the ion, Qion is the charge on the ion and Eðx; y; z; tÞ is the time-varying electric field. In the present simulations, damping of the ion was neglected. The above equation has no closed form solution. The equation is solved by the Runge-Kutta fourth order method [78] to obtain the ion trajectory. The initial position and velocity of the ion needs to be specified. 3.3. Pseudopotential well The pseudopotential well concept was used in this study to evaluate the strength with which the ions are trapped in the DC and RF fields. The effective potential was calculated by the following formula [79].

U ¼ UDC þ

2 Qion

! E RF ; 2 4mion u

(10)

where U is the pseudopotential, UDC is the potential at a point in space due to VDC (or Vapplied ), Qion is the charge on the ion of mass !  mion , u is the radial frequency of VRF and  E RF  is the effective peak

!

electric field along the x direction due to VRF . The E RF is calculated as the Euclidean (L2 ) norm of the peak field components [80]. The effective force field Eeff can be obtained from U by using a finite difference approximation. The pseudopotential can be found at any point in space. A potential well is formed above the trap surface due to the interaction of VDC (or Vapplied ) and VRF , called as pseudopotential well. Any ion, if left inside the well, is pushed towards the minimum of the well. This minimum is the equilibrium point where the ion stabilizes. The height at which the ion stabilizes defines the trapping height of

The results of simulations for the three geometries shown in Figs. 1e3 are presented here. In all the simulations, the positive ions considered were all singly charged and the frequency of VRF was kept at 1 MHz. The ion motion in all the directions is shown for a duration of 250 ms. However, the Fast Fourier Transform (FFT) [78] was computed by considering the ion motion for a longer duration of 5 ms. 4.1. Test Ion Trap Geometry A study of the Test Ion Trap Geometry was carried out to understand the different aspects of ion trapping with DC and RF potentials in single-PCB geometries. This includes studies on the roles of DC and RF potentials for trapping ions as well as the fields experienced by the trapped ions. These studies will provide insights into how fields need to be tailored in planar trap structures for them to be useful as mass analyzers. First, it was confirmed that the Test Ion Trap Geometry design is indeed capable of trapping ions in the presence of DC and RF potentials. Then, the importance of the DC potential in the trapping process was established. Further, it was showed that ions of different masses are trapped at different heights; this difference in height occurs due to the difference in zero crossings of DC and RF potentials along the z direction. Finally, it was shown that at these different heights, the fields experienced by the ions are different. 4.1.1. Trapping of ions The first simulation that was carried out demonstrated that the Test Ion Trap Geometry can indeed trap ions when appropriate DC and RF potentials are applied on the respective electrodes. The results of one such simulation for a given combination of VDC and VRF , which demonstrates stable ion motion in the three directions, is given in Fig. 7. Fig. 7 presents the motion of a singly charged ion of mass 78 u in the Test Ion Trap Geometry. For these simulations, VRF ¼ 40 V0p and VDC ¼ 5 V. The initial position of the ion in three directions was arbitrarily fixed at x ¼ 3 mm, y ¼ 1 mm and z ¼ 4 mm. The initial velocities were fixed at 0 m s1 in all the three directions. Fig. 7a, 7b and 7c show the motion of the ion in the x, y and z

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Fig. 7. Ion motion in Test Ion Trap Geometry along (a) x, (b) y and (c) z directions.

Table 1 Frequency and Amplitude of ion motion along x direction calculated by Harminv.

Fig. 8. FFT of ion motion along x direction in the Test Ion Trap Geometry.

directions respectively. The horizontal and vertical axes in the figures represent the time and amplitude of ion motion respectively. It is clear from these figures that the motion of the ion is bounded in

Frequency (Hz)

Amplitude (m)

20242.8 60723.0 30214.4 101146.0 979755.0 9158.12

1.48591  1.35732  1.29218  4.47317  3.49678  1.00772 

103 105 105 106 107 107

Ratio 1.0 9.1346  8.6962  3.0104  2.3533  6.7818 

103 103 103 104 105

all three directions, indicating that under these operating conditions, the ion is trapped for the duration of simulation (250 ms). The frequency spectrum of the x direction ion oscillation (shown in Fig. 7a) was obtained using FFT and is shown in Fig. 8. The peak value occurred at 20.2 kHz. Additionally, the frequency analysis of motion (shown in Fig. 7a) was carried out using the Harminv program [81,82]. The results of this computation are tabulated in Table 1. The various frequency components are arranged in decreasing order of their amplitude. The dominant oscillation frequency occurred at 20.24 kHz with an amplitude of 1.48 mm. The ratios of the amplitudes of different frequencies with respect to the amplitude of the dominant frequency are also tabulated. It can be seen that other frequency components, though present, have amplitudes which are smaller than two orders of magnitude compared to the dominant peak.

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An analysis of the ion motion in the y and z directions shown in Fig. 7b and c, respectively, not presented here for the sake of brevity, indicated that the motion in these two directions is anharmonic. 4.1.2. Importance of DC potential for trapping ions In the previous section, it was shown that the Test Ion Trap Geometry can trap ions for a given combination of VDC and VRF . In this section, the importance of the DC potential in trapping ions is considered. These simulations use the pseudopotential concept that was discussed in Section 3.3. Fig. 9a and b show the pseudopotential curves obtained for VRF alone and for VDC and VRF combination, respectively. These plots were obtained for a singly charged ion of mass 78 u at a height of z ¼ 4.15 mm above the trap surface. For these simulations, VRF ¼ 40 V0p and VDC ¼ 5 V. The role of the DC potential is very evident from Fig. 9b. In the absence of DC potential (Fig. 9a), the well depth is very small. In contrast, on application of a small DC potential (Fig. 9b), the pseudopotential plot changes dramatically and well depth increases by several orders of magnitude. In connection with use of DC potential in planar traps, the following two observations can be made. First, as has been seen, a small DC potential provides a larger trapping effect as compared to a large RF potential. This is especially important in PCB trap designs, where the electrodes are closely placed and the use of large potentials is not recommended. Furthermore, the focus is on the axial (x) direction where the DC potential plays a major role in trapping ions with marginal effect due to the RF potential. Second, the trapping effects of the DC and RF fields are qualitatively different. While the RF provides an inherent trapping field along the x direction, the DC field provides trapping along one direction (in the x direction) and destabilization in other directions (in y and z directions). The ion tends to destabilize easily in the perpendicular z direction due to weak confinement. This situation is similar to what exists in the LIT [62,63]. Based on the above discussion, it is seen that use of the DC potential (that confines ions along the axial direction) is necessary for trapping ions. 4.1.3. Variation in trapping heights of ions of different masses In this section, it is shown that the DC potential also plays a role in determining the height at which ions of different masses are

trapped above the trap surface. These studies also use the concept of the pseudopotential as discussed in Section 3.3. Fig. 10 shows the pseudopotential plots and variation of trapping height of singly charged ions of four different masses, viz., 78 u, 150 u, 200 u and 300 u. For these simulations VRF ¼ 40 V0p and VDC ¼ 5 V. The initial position of the ion in two directions was fixed at x ¼ 0 mm and y ¼ 0 mm. The initial velocities were fixed at 0 ms1 in all three directions. Fig. 10a shows the pseudopotential plots for the four masses. The plots indicate, from the minima of each curve, that ions of different masses experience different pseudopotentials with the lower masses experiencing higher pseudopotentials. Fig. 10b shows the variation in trapping height of the ion with mass of the ion. This plot was made by taking the z value corresponding to the minima of the pseudopotential curve (which corresponds to the trapping height of the ion) for each mass from Fig. 10a. The plot indicates that the trapping height decreases with increasing mass of the ion, implying that heavier masses are trapped closer to the plane of the trap compared to lighter masses. When the DC and RF potentials are simultaneously present, the ion settles at the equilibrium point determined by competing forces due to the DC and RF electric fields. To understand the settling of different masses at different heights, it is to be noted that while the force due to DC field depends only on charge and not mass of the ion, the force due to RF field depends on both charge and mass of the ion and the force is weaker for the heavier ion [51] (Fig. 10b). An undesirable consequence of different ions settling at different heights is that they experience different fields. The field components in the x direction, Ex and Ez , at different heights (indicated in Fig. 10b) are plotted in Fig. 11. For generating these plots, VRF was kept at 0 V0p because the contribution of the RF potential to the fields in the x direction is insignificant as seen in Fig. 9; only the DC potential was used and VDC was kept at 5 V. Fig. 11a shows the plot of the field Ex . It is seen that the Ex of the four ions overlap and are linear only very close to x ¼ 0. Fig. 11b shows the plot of the field Ez . The plot indicates that there is no overlap of the fields of the four masses and each mass experiences a different field. Hence, it is seen that in the Test Ion Trap Geometry, ions of different masses settle at different heights. It is also seen that at different heights, the fields are non-zero and different. The fact that ions of different masses experience different fields is not a desirable feature for a mass analyzer.

Fig. 9. The pseudopotential curves along x direction in the Test Ion Trap Geometry for (a) VRF alone and (b) both VDC and VRF .

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Fig. 10. Variation of (a) pseudopotential and (b) trapping height of ion, with mass of ion in the Test Ion Trap Geometry.

Fig. 11. The variation of field components (a) Ex and (b) Ez , along x direction in the Test Ion Trap Geometry.

Fig. 12. Field variation in the z direction for (a) unit RF potential and (b) unit DC potential in the Test Ion Trap Geometry.

9

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4.1.4. Field zero crossing heights along z axis To understand why ions of different masses settle at different heights, field zero crossings of the DC and RF potentials in the z direction were studied. Fig. 12a shows the variation of field with height, indicating zero crossing for the RF field. It was obtained by fixing a unit potential to the RF electrodes and ground potential to all other electrodes. Fig. 12b shows the variation of field with height, indicating zero crossing for the DC field. It was obtained by fixing a unit potential to the DC electrodes and ground potential to all other electrodes. The figures indicate that the field zero crossing points occur at different heights; for the RF field, this occurs at z ¼ 4.15 mm and for the DC field, this occurs at z ¼ 7.92 mm mm. Since these curves were plotted for unit potentials, the scaling of potentials does not alter the zero crossing point for a given trap geometry. In the absence of the DC potential, the ions were trapped at the RF field zero crossing. However, the DC field zero crossing point was above that of RF and the DC field is destabilizing in the z direction. Consequently, the ions would be pushed down by the DC field, resulting in a change in settling height of the ions. This effect is larger for heavier ions where the trapping force due to RF is weaker. Based on the above studies, the desired characteristics of fields in a planar ion trap for it to be used as a mass analyzer are presented. 4.2. Desired characteristics of fields in a planar ion trap It is first necessary to determine the trapping height from the zero crossing of the RF field on the z axis as outlined in Section 4.1.4. It is at this height that the ion oscillates harmonically along the x direction. For this to be achieved, the DC field in planar traps must have the following two characteristics along the x direction at the trapping height. First, the field component Ex must be proportional to x within the defined trapping region. Beyond this, in the computations, the field was made to fall to zero by the end of the trap. Second, the lateral field components Ey and Ez along x direction must be zero throughout the region of the trap. The second condition (Ey ¼ 0 and Ez ¼ 0) ensures that the field zero crossings of both DC and RF fields coincide on the z axis. These conditions are shown in Fig. 13a on a unit scale. The field component Ex varies linearly with displacement from the centre along the x direction in the trapping plane and decreases instantly to zero thereafter. The field component Ez along the x direction is

zero in the trapping region. In the One-Sheet Ion Trap Geometry, the field component Ey ¼ 0 because of symmetry in the structure and consequently, the conditions for only Ex and Ez need to be imposed. In the case of the Two-Sheet Ion Trap Geometry, both Ey ¼ 0 and Ez ¼ 0 because of symmetry in the structure and consequently, only the condition for Ex needs to be imposed. From the practical point of view, the abrupt fall of Ex to zero (as shown in Fig. 13a) may lead to numerical difficulties. So, the specification shown in Fig. 13b was used, in which Ex drops gradually to zero from its peak over a distance. The field conditions described above were imposed on the respective geometries by using a least square method [77] and this is referred to as tailoring of fields. Next, the method by which fields are tailored in the one-sheet ion trap geometry is described. Fig. 14a shows the side view of the One-Sheet Ion Trap Geometry with an imaginary line parallel to the x axis located at a height z0 , lying within the geometrical bounds of the trap. z0 represents the field zero point along the z direction due to VRF . The field and field zero point (z0 ) due to VRF were obtained from the surface charge distribution on the trap in the manner discussed in Section 3.1.3. The electrode placement was symmetrical about the origin and so was the electrode excitation. This produces a symmetrical field distribution along the x direction. So, for computational ease, only the line along the positive x axis was considered. The line was divided into equal segments by m points having the same spacing of Dx. Thus, the distance along the x axis from the origin in steps of Dx is given as

x ¼ ði  1ÞDx;

(11)

where the index variable i represents a point on an imaginary line and i ¼ 1, 2, / m. Fig. 14b shows the top view of the region around the central split electrode, marked with a dashed rectangle in Fig. 14a. Also shown in the figure is the labelling of segments of the split electrode. The central electrode was split into 2n þ 1 odd number of identical electrode segments, where n represents the number of pairs of segmented electrodes. For the One-Sheet Ion Trap Geometry, n was taken as 10 pairs. The central segment of the split electrode was grounded and assigned as the 0th electrode. All other segments were placed and numbered symmetrically about the 0th electrode from 1 to n. These 2n electrode segments were applied with DC

Fig. 13. The variation on unit scale along x direction in planar traps, of (a) ideal fields Ex and Ez and (b) practical field Ex .

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11

Fig. 14. The One-Sheet Ion Trap Geometry. (a) Side view showing imaginary line and (b) top view of the region marked with dashed rectangle. Indicated in the figures are the axes.

potentials. The segments assigned the same number were applied with the same DC potential, forming a split electrode segment pair. The method adopted to obtain the DC potentials to be applied on the central electrode segments is now considered. First, the z0 was determined. Then, the DC field zero point along z direction was enforced to lie on z0 . The index variable j represents a split electrode segment pair. A unit potential is applied on the jth electrode segment pair while all other electrodes in the trap are grounded. The field components Ex and Ez along the x direction at m points were computed and gathered in a transposed array as



T Ej ¼ Ex1;j Ex2;j / Exi;j / Exm;j Ez1;j Ez2;j / Ezi;j / Ezm;j : (12) Similarly, the field components at m points on the imaginary line for all the n electrode segment pairs were obtained and combined together in an array as

A ¼



     E1 E2 /Ej /En

(13)

where j ¼ 1, 2, / n. The number of points m on the imaginary line was chosen to be larger than the number of electrode segment pairs n because an overdetermined system of equations can be solved using a least square method [83,84]. The actual fields were in proportion to the applied DC potentials on each electrode segment pair. These applied DC potentials were unknown and had to be calculated. The unknown DC potentials on n electrode pairs are represented as an n-component vector

Vapplied ¼



V1

V2

/

Vj

/

Vn

T

:

(14)

The following conditions were imposed on the field components Ex and Ez at points on the imaginary line: Ex varies linearly to a distance given by m1 points, where m1 < m and becomes zero after (from m1 þ 1 to m). These conditions, on the One-Sheet Ion Trap Geometry at height z0 in the x direction, are represented as

Ex ðiÞ ¼ hði  1ÞDx

and

Ez ¼ 0 ;

(15)

A2mn Vapplied ðn1Þ ¼ b2m1 ;

and is solved to find Vapplied using the method of least squares [77]     for minimizing the Euclidean norm b  AVapplied . It should be noted that the application of Vapplied to the split electrodes produces a linear field Eðx;y;zÞ. As seen above, changing Vapplied to aVapplied changes the field from Eðx; y; zÞ to aEðx;y;zÞ, thus preserving the linear variation of the field along the x axis. This is referred to as scaling in this study.

4.3. Frequency estimation of ion motion In this section, an analytical expression for finding the oscillation frequency of ion motion in the One-Sheet Ion Trap Geometry and Two-Sheet Ion Trap Geometry is obtained. It was shown in Fig. 9 as a comparison that the DC field has a stronger effect than the RF field on ion motion in the axial direction. Hence, ion motion along the axial direction is largely governed by DC fields. Further, DC fields along the axial direction in the two planar ion trap geometries were tailored by the method discussed in Section 4.2. After tailoring the DC fields, the field component Ex varied linearly while the field components Ey and Ez were zero along the x direction. The force on an ion along the axial (x) direction can be obtained from Eq. (9) as

mion

d2 x ¼ Qion Ex ; dt 2

Ex ¼ hx;

(19)

where mion is the mass of the ion, Qion is the charge on the ion and h is the slope of Ex . Substituting the value of the field component Ex from Eq. (19) in Eq. (18),

mion

b ¼ ½Ex ð1Þ Ex ð2Þ / Ex ðiÞ / Ex ðm1 Þ / 0 0 0 / 0 / 0T :

k ¼  hQion :

The field component Ex ð1Þ at the centre of the trapping region at height z0 is zero, from Eq. (15). A system of linear equations is formed as

(18)

and from Eqs. (11) and (15)

where i ¼ 1, 2, / m1 and h is the slope of Ex . The h provides the desired linear field variation of Ex in the x direction. The resultant field components Ex and Ez at m points on the imaginary line are computed from Eq. (15) and gathered in a transposed array as

(16)

(17)

d2 x ¼ hQion x ¼ kx; dt 2

(20)

where k is the spring constant of an equivalent spring, given as

(21)

It is to be noted that the signs of Qion and h are opposite to make k positive. If this condition is not satisfied, the ion will not be trapped. Eq. (20) represents a simple harmonic motion whose frequency is given as

12

1 f ¼ 2p

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sffiffiffiffiffiffiffiffiffiffi k : mion

direction is found to be same as that due to VRF (4.2 mm).

(22)

It should be noted that by scaling the applied DC potentials on all the electrode segment pairs (represented by the vector Vapplied in Eq. (14)) by a factor a, the slope scales by the same factor a and pffiffiffi therefore, the frequency scales by factor a as can be deduced from Eq. (22). In the next two sections, the study results on the One-Sheet Ion Trap Geometry and Two-Sheet Ion Trap Geometry are presented. 4.4. One-sheet ion trap geometry The parameters of the One-Sheet Ion Trap Geometry were discussed in Section 2.2. The central electrode was arbitrarily split into 21 segments. The Vapplied was computed in the manner discussed in Section 4.2 to obtain the desired fields. It is to be remembered that because of symmetry in the structure, the field component Ey is zero. The conditions that need to be imposed pertain to the field components Ex and Ez . 4.4.1. Field variation in the one-sheet ion trap geometry At the start of tailoring fields in the One-Sheet Ion Trap Geometry, the height at which ions will settle in the presence of the RF potential alone (z0 ) is determined. This is done by putting a unit potential on the RF electrodes and grounding all other electrodes. In the present simulations, the height was found to be z ¼ 4.2 mm. The tailoring of the field components Ex and Ez was then carried out by taking the height of the trapping plane to be 4.2 mm. The Vapplied was obtained by imposing the required conditions in the manner outlined in Section 4.2, at z ¼ 4.2 mm. The field component Ex was tailored to be linear within the bounds x ¼ 5 mme5 mm, then drop linearly to zero from x ¼ 5 mm to 10 mm and x ¼ 5 mme10 mm and continue to be zero for x ¼ 10 mm to 13 mm and x ¼ 10 mme13 mm. Table 2 lists the computed Vapplied on the different segmented electrode pairs; the rounded-off values for Vapplied on the corresponding electrode pairs are also given. These round off potential values are used in our simulations. Fig. 15 shows the comparison of field variation along the x direction, between the computed and rounded-off DC potential values. The Figs. 15a and b show the variation of fields Ex and Ez respectively; it is seen that the field variations due to computed and rounded-off DC potential values match closely. The figures indicate that Ex varies linearly and Ez is close to zero, over a distance along the x direction. The field zero crossing due to rounded-off Vapplied values along z Table 2 Potentials to be applied on segmented electrodes. Electrode

Vapplied (V) Computed

Round off

0 1 2 3 4 5 6 7 8 9 10

0.0 0.0496 0.5526 1.2419 2.0389 5.1558 8.6654 2.6151 10.0810 16.0569 5.7784

0.0 0.0 0.6 1.2 2.0 5.2 8.7 2.6 10.1 16.1 5.8

4.4.2. Trapping of ions Fig. 16 presents the motion of a singly charged ion of mass 78 u in the One Sheet Ion Trap Geometry (Fig. 2). For these simulations, the Vapplied values were the rounded-off values listed in Table 2 and VRF ¼ 200 V0p . The frequency of VRF was 1 MHz. The initial position of the ion in three directions was arbitrarily fixed at x ¼ 2.5 mm, y ¼ 1 mm and z ¼ 4.2 mm. The initial velocities were fixed at 0 ms1 in all three directions. The simulation was carried out for a time period of 250 ms. The Fig. 16a, 16b and 16c show the motion of the ion in the x, y and z directions respectively. It is seen that the motion of the ion is bounded in all three directions, indicating that the ion is trapped under these operating conditions. The FFT [78] of the x direction oscillation of the ion (shown in Fig. 16a) is presented in Fig. 17. The peak value of the figure (shown with a line) was at 15.2 kHz, with all other peaks being very small. Thus, it can be concluded that the ion motion is dominantly sinusoidal along the x direction. Additionally, the frequency analysis of motion (shown in Fig. 16a) was carried out using the Harminv program [81,82]. The results of this computation are tabulated in Table 3. The various frequency components and the corresponding amplitudes are arranged in decreasing order of amplitude. The dominant oscillation frequency occurred at 15.26 kHz with an amplitude of 1.25 mm. The ratios of the amplitudes of different frequencies with respect to the amplitude of the dominant frequency are also presented in Table 3. It can be seen that the amplitudes of other frequencies are less than 0.8% of the amplitude of the dominant frequency. In order to compute the frequency of ion motion from Eq. (22), the values for slope h, charge of ion Qion and mass of ion mion have to be substituted. In the simulations, an arbitrarily value was chosen for h ¼ 1.0  104 V m2. For a singly charged ion of mass 78 u, the frequency of ion motion was computed (from Eq. (22)) to be 17.7 kHz. The frequencies obtained from FFT (15 kHz) and Harminv (15.26 kHz) were different from that obtained from Eq. (22). This difference may be because the field Ex was not as linear and Ez was not zero in the One-Sheet Ion Trap Geometry as in the ideal case. 4.5. Two-sheet ion trap geometry The parameters of the Two-Sheet Ion Trap Geometry were discussed in Section 2.3. Fig. 18 shows the side view of this geometry, portraying the different electrodes on one PCB. The figure also shows the imaginary line along the positive x axis, divided into equal segments by m points with the same spacing Dx. Here, the central electrode on each sheet was arbitrarily split into 25 segments. The DC potentials Vapplied , to be applied on different segments of the split electrode on each sheet were computed in the manner discussed in Section 4.2 to obtain the desired fields. It is to be remembered that because of symmetry in structure, the field components Ey and Ez are zero. The only condition that needs to be imposed pertains to the field component Ex . It is to be noted that the central segment of the split electrode, that is the 0th electrode, is not grounded. 4.5.1. Field variation in the two-sheet ion trap geometry The RF field zero crossing in the Two-Sheet Ion Trap Geometry lies in the middle of the sheets because of the symmetry in structure. The Vapplied on each sheet was obtained by imposing the required conditions of field in the manner outlined in Section 4.2, along the x axis in between the two sheets. The field component Ex was tailored to be linear within the bounds x ¼ 10 mme10 mm,

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Fig. 15. Variation of field components (a) Ex and (b) Ez , in the x direction. Each figure shows the comparison between the plots for computed and rounded-off values of Vapplied .

Fig. 16. Ion motion in One-Sheet Ion Trap Geometry along (a) x, (b) y and (c) z directions.

beyond which the field was not considered. Table 4 lists the Vapplied that have been computed; the roundedoff values for Vapplied on the corresponding electrodes are also

given. These round off potential values are used in simulations. Fig. 19 shows the comparison of field variation along the x direction, between the computed and rounded-off DC potential

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Fig. 17. FFT of ion motion along x direction in One-Sheet Ion Trap Geometry.

Table 3 Frequency and Amplitude of ion motion along x direction calculated by Harminv. Frequency (Hz)

Amplitude (m)

15263.9 8502.1 45791.2 38997.3 38585.8

1.2482  9.8469  6.8444  4.8509  1.0563 

103 106 106 106 106

Ratio 1.0 7.8887  5.4832  3.8862  8.4626 

103 103 103 104

Fig. 19. The variation of field component Ex along x direction. The figure shows the comparison between the plot for computed and round off values of Vapplied .

values. It is seen that the field variations due to the computed and rounded-off DC potential values match closely. The figure indicates that Ex varies linearly over a distance along the x direction. However not shown, Ez along x direction is zero in the middle of the structure. Due to symmetry in the electrode design in the two-sheet ion trap geometry, the field zero crossings along the z axis due to Vapplied and VRF coincide in the middle of the structure at the origin. As an aside, to demonstrate that the method outlined in Section 4.2 can also be used to obtain mildly superlinear fields, consider the field, Ex , in the Two-Sheet Ion Trap Geometry to have the form

  x 2  ; Ex ¼ h x 1 þ l a

Fig. 18. Side view of the Two-Sheet Ion Trap Geometry.

Table 4 Potential to be applied on segmented electrodes. Electrode

0 1 2 3 4 5 6 7 8 9 10 11 12

Vapplied (V) Computed

Round off

2.8334 2.8264 2.8052 2.7698 2.7201 2.6561 2.5775 2.4836 2.3758 2.2413 2.1490 1.7436 2.1849

2.8 2.8 2.8 2.8 2.7 2.7 2.6 2.5 2.4 2.2 2.1 1.7 2.2

(23)

where h is the slope of the linear part of Ex (Eq. (19)), l is a measure of the cubic non-linearity and a is half the length of the interval over which a linear field variation is desired. l should be positive for a superlinear variation. For generating the curves in Fig. 20, h is arbitrarily chosen as 4.0  103 V m2 and a ¼ 10 mm in Eq. (23). Fig. 20a present the curves obtained from Eq. (23), corresponding to the contribution of the linear term (dashed line) and the desired superlinear field for l ¼ 0.1 (solid line). Using the least square method (Section 4.2), the required electrode potentials that were obtained are listed in Table 5. When these potentials are applied on the relevant electrodes, the corresponding tailored field obtained is shown as a dotted line in Fig. 20b. Also shown in the same figure with the solid line is the field that was desired. An inspection of the two curves show that the match is quite good, indicating that the technique outlined in this study can also provide mildly superlinear fields. 4.5.2. Trapping of ions Fig. 21 presents the motion of a singly charged ion of mass 78 u in the Two-Sheet Ion Trap Geometry. For these simulations, the Vapplied values were the rounded-off values listed in Table 4 and VRF ¼ 100 V0p . The frequency of VRF was 1 MHz. The initial position of the ion in three directions was fixed in the middle of the two PCBs at x ¼ 10 mm, y ¼ 1 mm and z ¼ 0 mm. The initial velocities were fixed at 0 ms-1 in all three directions. The simulation was carried out for a time period of 250 ms.

N. Goyal, A.K. Mohanty / International Journal of Mass Spectrometry 446 (2019) 116218

15

Fig. 20. Variation of field with distance (a) contribution of the linear term and the desired superlinear field for l ¼ 0.1 (b) desired superlinear field for l ¼ 0.1 and superlinear field obtained by least square method.

Table 5 Potential to be applied on segmented electrodes in the Two-Sheet Ion Trap Geometry to obtain superlinear field (l ¼ 0.1). Electrode

Vapplied (V)

0 1 2 3 4 5 6 7 8 9 10 11 12

3.4165 3.4100 3.3905 3.3577 3.3111 3.2501 3.1740 3.0809 2.9723 2.8285 2.7485 2.2197 2.9056

The Fig. 21a, 21b and 21c show motion of the ion in the x, y and z directions respectively. It is seen that the motion of the ion is bounded in all three directions, indicating that the ion is trapped under these operating conditions. The frequency spectrum of the ion oscillation, shown in Fig. 21a, was obtained by FFT [78]. Fig. 22 shows the graph of frequency with amplitude. The maximum amplitude occurred at 11.2 kHz, with all other peaks being very small. Here too, it can be concluded that the ion motion is dominantly sinusoidal along the x direction. The frequency analysis of motion, shown in Fig. 21a, was performed using the Harminv program. The results of this computation are tabulated in Table 6. The various frequency components and the corresponding amplitudes are arranged in decreasing order of amplitude. The dominant oscillation frequency occurred at 11.18 kHz with an amplitude of 4.99 mm. The ratios of the amplitudes of different frequencies with respect to the amplitude of the dominant frequency are presented in Table 6. It can be seen that the amplitudes of other frequencies are less than 0.2% of the amplitude of the dominant frequency. In order to compute the frequency of motion of a singly charged ion of mass 78 u from Eq. (22), an arbitrarily chosen value of the slope h ¼ 4:0  103 V m2 was substituted in the simulations. This yielded the frequency as 11.19 kHz, which is similar to those obtained from FFT (11.2 kHz) and Harminv (11.18 kHz). The agreement in frequencies is much better in the case of the Two-Sheet Ion Trap Geometry as compared to the One-Sheet Ion Trap Geometry. This is expected because Ez is automatically zero in the Two-Sheet

Ion Trap Geometry and the method of least squares focuses only on Ex . 5. Concluding remarks In this paper, a simulation study on tailoring the fields in planar ion trap geometries for mass analysis has been presented. The motivation was to obtain linear fields in the proposed traps, similar to the fields in the Paul Trap mass analyzer. Two different planar trap geometries were considered: the first was a One-Sheet Ion Trap Geometry in which ions are trapped off-plane and the second was a Two-Sheet Ion Trap Geometry in which ions are trapped in between. Both DC and RF potentials were used for trapping the ions. Initially, a trap design referred to as the test Ion Trap Geometry was taken up to understand the conditions under which ions are trapped in planar traps. The results of this study was then used for tailoring fields in the two proposed ion trap geometries, namely, the One-Sheet Ion Trap Geometry and the Two-Sheet Ion Trap Geometry. The One-Sheet Ion Trap Geometry consisted of five metal electrodes on a 30 mm by 30 mm Teflon base. The central electrode had DC potential while the two outer electrodes were grounded and the two inner electrodes had RF potential applied on them. In this trap, ions are trapped off-plane. In the case of the TwoSheet Ion Trap Geometry, each sheet consisted of three metal electrodes on a 40 mm by 20 mm Teflon base. The central electrode had DC potential, while the other two electrodes were given out-ofphase RF potential. In this trap, the two sheets faced each other and trapping occurred in between the sheets. In the proposed two traps, the central electrode was segmented and the appropriate DC potential was applied on each segment. The applied DC potentials required to obtain the desired linear fields pattern were computed using a least square method. The results presented for the One-Sheet Ion Trap Geometry and the Two-Sheet Ion Trap Geometry were obtained by rounding-off the computed DC potentials to one decimal place to have practical values. However, better results can be obtained by using the precise computed DC potential values. In both these traps, it has been demonstrated that by tailoring the fields, near-linear fields were obtained. Further, it was shown that the stored ion has a sinusoidal motion that can be used for mass analysis. The authors believe that this study will be helpful in the context of using planar traps for mass analysis. However, before this happens, several other studies need to be done. These include the study

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Fig. 21. Ion motion in Two-Sheet Ion Trap Geometry along (a) x, (b) y and (c) z directions.

Table 6 Frequency and Amplitude of ion motion along x direction calculated by Harminv. Frequency (Hz)

Amplitude (m)

11184.8 33551.5 55920.9 144566.0 99635.0

4.9937  9.5034  2.2219  2.7856  1.9661 

103 106 106 107 107

Ratio 1.0 1.9031  4.4494  5.5782  3.9372 

103 104 105 105

with scaling of the trap dimensions. Acknowledgements We thank Professors S. Sampath and A.G. Menon for discussions during the preparation of this manuscript. We thank Ms. Geethanjali Monto for copy-editing the manuscript. We also thank an anonymous reviewer for helpful comments on the original manuscript. Fig. 22. FFT of ion motion along the x direction in Two-Sheet Ion Trap Geometry.

of stability plots and evaluation of multipoles to estimate the secular frequency of ion motion. Further, the effects can be studied

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