B American Society for Mass Spectrometry, 2013
J. Am. Soc. Mass Spectrom. (2014) 25:498Y508 DOI: 10.1007/s13361-013-0784-9
RESEARCH ARTICLE
Space Charge Induced Nonlinear Effects in Quadrupole Ion Traps Dan Guo,1 Yuzhuo Wang,1 Xingchuang Xiong,2 Hua Zhang,3 Xiaohua Zhang,3 Tao Yuan,4 Xiang Fang,2 Wei Xu1,5 1
Department of Biomedical Engineering, Beijing Institute of Technology, Beijing, China National Institute of Metrology, Beijing, China 3 Beijing Purkinje General Instrument Co., Ltd, Beijing, China 4 KunShan Innowave Communication Technology Co., Ltd, Jiangshu, China 5 School of Life Science, Beijing Institute of Technology, Haidian, Beijing, 100081, China 2
Abstract. A theoretical method was proposed in this work to study space charge effects in quadrupole ion traps, including ion trapping, ion motion frequency shift, and nonlinear effects on ion trajectories. The spatial distributions of ion clouds within quadrupole ion traps were first modeled for both 3D and linear ion traps. It is found that the electric field generated by space charge can be expressed as a summation of even-order fields, such as quadrupole field, octopole field, etc. Ion trajectories were then solved using the harmonic balance method. Similar to highorder field effects, space charge will result in an “ocean wave” shape nonlinear resonance curve for an ion under a dipolar excitation. However, the nonlinear resonance curve will be totally shifted to lower frequencies and bend towards ion secular frequency as ion motion amplitude increases, which is just the opposite effect of any even-order field. Based on theoretical derivations, methods to reduce space charge effects were proposed. Key words: Space charge, Quadrupole ion trap, Nonlinear resonance curve, Harmonic balance method Received: 27 August 2013/Revised: 28 October 2013/Accepted: 9 November 2013/Published online: 3 January 2014
Introduction
C
ombined with collision induced dissociation (CID) [1, 2] and electron transfer dissociation (ETD) [3–5] techniques, quadrupole ion traps have been widely used in mass spectrometry (MS) systems as mass analyzers and/or ion trapping reaction devices. As an ion manipulation device, the performance of a quadrupole ion trap is greatly influenced by space charge effects. With increasing number of ions trapped inside a quadrupole ion trap, Coulomb force between ions will modify ion trajectory and result in chemical mass shift, resolution degradation, and etc. [6–11]. Great efforts have been made to explore and minimize space charge effects in quadrupole ion traps.
Dan Guo and Yuzhuo Wang contributed equally to this work. Electronic supplementary material The online version of this article (doi:10.1007/s13361-013-0784-9) contains supplementary material, which is available to authorized users. Correspondence to: Wei Xu; e-mail: [email protected]
To understand space charge effects, simulation and experimental researches have been carried out. Due to buffer gas cooling effect, an ion cloud will reach thermal equilibrium inside a quadrupole ion trap [12, 13]. Laser tomography experiments [14–16] and simulations [8, 17, 18] have been performed to study the spatial distributions of ions within 3D quadrupole ion traps, which showed that ion clouds have close to Gaussian distributions along the radial and axial directions under equilibrium. Simulation and experiments have also shown that space charge will cause chemical mass shift, limited dynamic range, peak covalence, and resolution degradation not only in quadrupole ion traps [7–11, 15, 16, 19] but also in Fourier transform ion cyclotron resonance (FT-ICR) cells [20–26], Orbitraps [27–29], and digital ion traps [30, 31]. On the other hand, Coulomb force between ions possessing opposite polarities has also been utilized to enhance ion trapping [32]. Experiments were also performed to investigate the best operating conditions in linear ion traps to minimize space charge induced mass shift and resolution degradation [11, 33]. Besides simulation and experiments, theoretical modeling would be a powerful tool to gain an in-depth understanding of space charge effects [6, 34–36].
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
Utilizing the classic Boltzmann distribution, ion spatial distributions within quadrupole and Penning ion traps have been calculated under thermal equilibrium conditions [13, 17, 18, 33, 37]. Ion trapping capacity of quadrupole ion guides (when used as ion trapping devices) and quadrupole ion traps were studied through balancing the effective rf trapping and space charge repulsion fields [9, 13, 37, 38]. Based on a relatively straightforward trapping volume calculation, the ratios of the maximum number of ions stored in a linear ion trap to that in a 3D ion trap were estimated [11, 39, 40]. The interactions between two ion clouds in FT-ICR cells have been explored by modeling an ion cloud with point charge, line charge, uniform charged disk, uniform charged cylinder, and uniform charged sphere, respectively [41–43]. However, because of the complexity of Coulomb interactions (for instance, ion motion equation will become nonlinear when including Coulomb forces), it still lacks effective methods of studying space charge effects on ion motion in quadrupole ion traps. In this work, theoretical models have been first proposed to study the ion cloud distribution within both 3D and linear ion traps. Furthermore, space charge effects on ion motion characteristics have been explored, including ion motional frequency shift and the nonlinear resonance curve. Starting from Boltzmann distribution, an ion cloud is found to have close to Gaussian distributions along the quadrupole field directions within both 3D and linear ion traps. In this study, the electrical field originated from space charge effects was expressed as a summation of high-order fields, and the harmonic balance method [44, 45] was then used to solve the ion motion equation. Space charge-induced ion frequency shift and nonlinear resonance curve were explored. With the knowledge obtained from space charge modeling, it is found that ion motion frequency shifts originated from Coulomb interaction could be reduced by operating at higher q values and higher ion cloud temperatures.
Experimental Ion Cloud Modeling To characterize space charge effects on ion motions in quadrupole ion traps, ion distributions within both 3D and linear ion traps were first investigated. When trapped inside a quadrupole ion trap, ions will form an ion cloud. With the pseudopotential approximation, the ion density distribution (or ion spatial distribution) in an ion trap meets Boltzmann distribution under thermal equilibrium [13], which can be written as:
nðx; y; zÞ ¼ N
∫
−q ϕ ðx; y; zÞ exp KT −q ϕ ðx; y; zÞ dxdydz exp KT
ð1Þ
499
where N is the total number of ions, q is the electric charge that an ion possesses, K is the Boltzmann constant, T is the temperature of the ion cloud and ϕ represents the electric potential within the ion trap. Since the denominator is a constant, Equation (1) can be simplified as: nðx; y; zÞ ¼ N 0 exp
−qϕ ðx; y; zÞ KT
ð2Þ
The electric potential ϕ can be written as the summation of rf (radio frequency) trapping potential (ϕI) and the potential induced by space charge (ϕS). ϕ ¼ ϕI þ ϕS
ð3Þ
To isolate space charge effects from high-order field effects, ion traps with pure quadrupole electric field were considered in this work. According to the pseudopotential approximation theory, the rf trapping potential in a 3D ion trap can be expressed as [46]: ϕI ¼
Dr 2 Dz 2 r þ 2z þC r20 z0
ð4Þ
in which r0 is the radius of the ring electrode, z0 is the distance from trap center to end cap and C is a constant. Dr and Dz represent the pseudopotential well depth along the radial (r) and axial (z) directions, and Dr ¼
mr20 Ω 2 q2 mz2 Ω 2 q2 a x þ x ; Dz ¼ 0 az þ z ð5Þ 2q 2 2 2q 2 2
where ax, az, and qx, qz are the dimensionless constants in the Mathieu Equation, Ω is the rf frequency. Since Dr/r02 may not equal to Dz/z02, the ion cloud within a 3D ion trap has an elliptical shape in most cases. If Dr/r02 = Dz/z02 or at a higher q values, an ion cloud could form a spherical shape [8]. For simplicity, a spherical ion cloud shape is assumed within the 3D ion trap in this study, and the rf trapping potential can be approximated as [13],
ϕI ¼
Dr 2 u þC r20
ð6Þ
where u is the distance from the point of interest to trap center. In a linear ion trap, the rf trapping potential in the x-y plane can be expressed as,
ϕI ¼
Dr 2 r þC r20
ð7Þ
500
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
where au is set to zero. Along the z-axis, a zero electric potential is assumed within the effective trapping length of a linear ion trap, and a uniform ion distribution is applied. With controlled number of ions, the potential induced by space charge is small comparing to the rf trapping potential, and the ion cloud distributions could be approximated as Gaussian distributions [13]. Substituting Equations (6) and (7) back to Equation (2), Equation (2) reduces to the form of a Gaussian distribution for ions with the same m/z (mass-tocharge) ratio. Therefore, in a 3D ion trap, ion density distribution could be expressed as, 2 N −u nðuÞ ¼ pffiffiffiffiffi 3 exp 2a2 2π a
ð8Þ
where a is the standard deviation (SD). In a linear ion trap, ion density distribution in the x-y plane could be written as,
nð r Þ ¼
2 ρl −r exp 2a2 π 2a2
ð9Þ
in which ρl is the ion line density, ρl = N/z0 and z0 is the effective length of the linear ion trap. With the ion distributions shown in Equations (8) and (9), the electric field (E) induced by space charge could be calculated using Gaussian theorem (details in Supplementary Information): 8
u > > u2 p ffiffi ffi qN Erf > − > > e 2a2 2a < E ðuÞ ¼ − pqN ffiffiffi 3D ion trap þ 4ε0 πu2 2 2aε0 π 3=2 u > 2
> > qρl −r > > 1 − exp linear ion trap :E ðrÞ ¼ 2πrε0 2a2 ð10Þ where Erf is the error function and ε0 is the permittivity of vacuum. The corresponding electric potential induced by space charge, ϕS can then be obtained by integrating the electric field (details in Supplementary Information). 8
u > > p ffiffi ffi NqErf > > > 2a
Z > ∞ > qρl −r2 > > dr linear ion trap 1− exp : ϕs ¼ 2πrε0 2a2 r ð11Þ
To obtain an exact ion distribution within an ion trap, an iterative process needs to be taken: solving the electric potential caused by the ion density distribution; substituting the electric potential into Equation (2) and a new ion density
distribution is computed; and repeating the above process numerically until two successive iteration results agree with each other. Both experimental measurements [14–16] and simulations [8, 17, 18] have shown that ion clouds have close to Gaussian distributions within a quadrupole ion trap. As a first order approximation, the above iteration was carried out once in this work to get an analytical solution as well as simplifying the calculation process. Substituting Equation (6), (7), (11) into Equation (2) and applying the second order Taylor expansion on the electric potential induced by space charge (ϕS), SD of the ion cloud distributions could be calculated, 8 1 −q Dr Q > > > 3D ion trap < − 2a2 ¼ KT r2 − pffiffiffi 3 3=2 12 2a π ε0 0 > 1 −q Dr ρl q > > − linear ion trap :− 2 ¼ 2a KT r20 8 a2 πε0 ð12Þ in which Q is the total charge inside the 3D ion trap, Q = qN. Thus rffiffiffi 3 2 pffiffiffi k π T ε0 ρ20 3 a¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=3 pffiffiffi 3 2Dr 2 q3 Qε20 ρ20 þ 18Dr 4 q6 Q2 ε40 ρ40 − 96Dr 3 k 3 π 3 q3 T 3 ε60 ρ60 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3 2Dr 2 q3 Qε20 ρ20 þ 18Dr 4 q6 Q2 ε40 ρ40 − 96Dr 3 k 3 π 3 q3 T 3 ε60 ρ60 1=3 p ffiffiffiffiffiffiffiffi pffiffiffi þ 3 144Dr π qε0
ð13Þ in a 3D ion trap, and ρ a¼ 0 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4kπT ε0 þ q2 ρl 2πqε0 Dr
ð14Þ
in a linear ion trap.
Ion Motion Modeling After knowing the ion cloud distribution and space charge induced electric field, space charge effects on ion motion could be studied. In this work, ion motion along the ejection axis was explored, for instance the z-axis in a 3D ion trap, the x or y-axis in a linear ion trap. Applying the pseudopotential approximation and considering space charge Coulomb force, ion motion inside an ion trap can be written as: u00 þ ω20 u ¼
q E m
ð15Þ
where u could be z in a 3D ion trap and x or y in a linear ion trap; ω0 is the secular frequency of the ion (angular
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
frequency); E is the electric field induced by space charge as shown in Equation (10). In order to solve Equation (15), the electric field E was expanded using an 11th order polynomial fitting method. E ðuÞ ¼ p1 u þ p3 u3 þ p5 u5 þ p7 u7 þ p9 u9 þ p11 u11
ð16Þ
where pn (n = 1, 3, 5, 7, 9, 11) are polynomial coefficients. The electric field is an odd function (electric potential an even function), since the Coulomb force between either positive or negative ions is always pushing ions away from the center. Space charge-induced electric field can be well represented by the 11th order polynomial fitting in both 3D and linear ion traps (details in Supplementary Information). Equation (16) indicates that the electric field induced by space charge can be treated as a summation of even-order fields. The corresponding relationship between high-order field components and the polynomial coefficients in Equation (16) can be found in Supplementary Information. Replacing E in Equation (15) with Equation (16), u00 þ ω20 u ¼ P1 u þ P3 u3 þ P5 u5 þ P7 u7 þ P9 u9 þ P11 u11
ð17Þ
where P ¼ mq p , Equation (17) can be solved directly by applying the harmonic balance method, which was introduced earlier in exploring high-order field effects in ion traps [44, 45]. Briefly, with the first-order approximation, the solution of Equation (17) can be written as u ¼ a0 þ a1 cosðωt Þ
ð18Þ
where a0 and a1 are coefficients of the zero-order and firstorder harmonic functions; ω is the ion motion angular frequency. Ion motion characteristics and trajectory can be calculated by substituting Equation (18) back into Equation (17) and balancing the coefficients of harmonic functions (details in Supplementary Information). With the presence of space charge Coulomb force, ion motion angular frequency will be modified as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3P3 a21 5P5 a41 35P7 a61 63P9 a81 231P11 a10 1 ω ¼ ω20 − P1 − − − − − 4 8 64 128 512
ð19Þ
During a typical ion ejection and detection process, AC (alternating current) dipolar excitation and buffer gas damping effects need to be considered. Modeling ion-
501
molecule collisions with a damping term (c), ion motion equation becomes 0 u00 þ cu þ ω20 u ¼ P1 u þ P3 u3 þ P5 u5 þ P7 u7 þ P9 u9 þ P11 u11 þ Pac cosðωac t Þ
ð20Þ
in which ωac is the angular frequency of the AC excitation signal; Pac accounts for the excitation strength, which was well defined in earlier works [47, 48]. Details about the calculations of Pac can be found in Supplementary Information). Following a similar method, the solution of Equation (20) can be written as u ¼ a0 þ a1 cosðωac t Þ þ a2 sinðωac t Þ
ð21Þ
The relationship among AC excitation frequency, ion motion frequency and amplitude (the nonlinear resonance curve) can be obtained after solving Equation (20) using the harmonic balance method (details in Supplementary Information). The 3D ion trap studied in this work has the dimensions of r0 = 5 mm, z0 = 3.535 mm; while the linear ion trap has the dimensions of x0 = y0 = 4 mm and z0 = 30 mm. The rf signals applied on both ion traps have a frequency of 1 MHz, and the rf voltage will be adjusted accordingly as qu was changed. Helium was used as the buffer gas with a pressure of 1 mTorr. Calculations were based on ions with m/z = 195 Da at a temperature of 300 K, unless otherwise specified.
Measurement of Errors In the derivation process, the pseudopotential approximation and the assumption that an ion cloud has a close to Gaussian distribution were used. The pseudopotential approximation has been widely used and tested to be accurate at low q values [46]. It has also been verified by theoretical [13], numerical [8, 17, 18], and experimental studies [14–16] that an ion cloud will have a close to Gaussian distribution with controlled number of ions. In this work, both of the above conditions were tried to be maintained in later calculations. Besides the two assumptions above, there are two major approximations: (1) the use of polynomial fitting to approximate the electric field induced by space charge; (2) the use of the harmonic balance method to solve the ion motion equation. The accuracy of this theoretical method is estimated by comparing the calculated frequency shifts with those obtained from a numerical method (details in Supplementary Information). Results show that the calculation error is within ~2 % for the 3D ion trap and ~6 % for the linear ion trap under typical operation conditions.
502
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
(b)
(a)
qz
0.1 0.4
0.2
0.32
104 105 105 106
N=5 N=1 N=5 N=1
normalized STD
104 mm-3)
0.05
0.5
(c)
9
Ion density (
0.6
Ion cloud size (normalized STD)
6
3
0.3 103
104
105
106
0.16
0.08
0.3 0.4
0.24
0 -1.0
107
-0.5
0.0
0.5
1.0
0
1
z/z0
N
2
3
Ion number
4
5 106
Figure 1. Ion density distributions in the 3D ion trap. (a) Normalized SD of ion clouds with different values of qz and ion abundance; (b) ion density distributions along the z-axis under different ion abundance; (c) Ion cloud size versus ion number. T = 300 K, r0 = 5 mm, z0 = 3.536 mm, m/z = 195 Da. As qz changes from 0.3 to 0.6, the rf voltage was swept from ~149 to 299 V0-p
cloud can be characterized using the SD (a). Calculated from Equation (13), Figure 1a plots the ion cloud size (normalized SD) versus the total ion number (N) and rf trapping condition (qz) in the 3D ion trap. Under the pseudopotential approximation, the ion trap has deeper trapping potential well depth at higher qz values, which results in tighter ion clouds. Figure 1b plots the ion distributions inside the ion trap at the same qz value but different ion numbers. As ion number increases, the ion cloud would expand rapidly with low ion numbers and slower with high ion numbers (Figure 1c), which agrees with simulation [8] and experimental results [15].
Results and Discussion Previous simulation and experimental results have shown that space charge would cause mass shift, resolution degradation and peak coalescence [7–11, 15, 16, 19]. Expressed as a summation of high-order fields as shown in Equation (16), space charge would also introduce nonlinear phenomena on ion motion characteristics.
3D Ion Trap Ion Cloud Size When cooled inside an ion trap, the size of an ion cloud has strong dependence on the rf trapping field and number of ions. While the Coulomb force pushes ions away from each other, the rf trapping field would compress ions towards the center of an ion trap. With a Gaussian distribution, the size of an ion
Ion Secular Frequency The effects of space charge on ion motion frequency were characterized by solving Equa-
(a)
0
N = 5 105 N = 1 10 6 N = 1.5 106
-5
qz = 0.36 qz = 0.46 qz = 0.56
-7
f-f0(KHz)
f-f0(KHz)
(b)
0
-10
-15
-14
-20
-21 0.0
0.2
0.4
z/z0
0.6
0.8
1.0
-25 0.0
0.2
0.4
0.6
0.8
1.0
z/z0
Figure 2. Ion secular frequency shifts with different ion abundance and qz = 0.46 (a); at different qz values and n = 1 × 106 (b). T = 300 K, r0 = 5 mm, z0 = 3.536 mm, m/z = 195 Da, q = 1 × e
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
tion (19) at different ion motion amplitudes (z), ion numbers (N), and qz values. In the calculation, a group of ions with m/ z = 195 Da was first placed in the 3D ion trap, then the motion of a single ion (m/z = 195 Da) was studied at different motional amplitudes. In general, ion motion frequency will decrease with the presence of Coulomb forces as shown in Figure 2, which agrees with simulation and experimental observations [7, 8, 10]. The space charge induced electric field (in Equation (10)) can be expressed in a polynomial form (Equation (16)), which can be seen as a summation of even-order fields (such as quadrupole, octopole, etc.). Furthermore, the Coulomb force will always push ions away from trap center, which resembles a net negative even-order field. Therefore, space charge effects will weaken the quadrupole field (P1 term in Equation (10)), as well as adding higher-order fields into the ion trap (P3, P5 … terms in Equation (10)). Mathematically, P1, P5, P9 … in Equation (16) are positive and P3, P7, P11 … are negative. In Equations (16) and (19), P1, P5, and P7 terms will be larger than P3, P7, and P11 terms, respectively. Thus, with weakened quadrupole field dominating, the modified ion motion frequency is always lower than the ion motion frequency without considering space charge effects.
While sharing similar features with high-order field effects, space charge effects are different from those arisen from any single high-order field component. Figure 2a plots the ion motion frequency shifts versus ion motion amplitudes at different ion numbers. The relative frequency shift from the ideal value (ion secular frequency without considering space charge effects f0) is larger at lower ion motion amplitudes and smaller at higher motion amplitudes, which are just the opposite effects caused by any even-order field [49, 50]. For instance, with an added octopole field, ion motion frequency will center at its secular frequency (f0) at low ion motion amplitudes, and shifts away from the secular frequency as ion motion amplitude increases [44, 51, 52]. A potential explanation is that with the pseudopotential approximation, ion motion within an ion trap can be treated as a harmonic oscillator. The characteristic frequency of a harmonic oscillator is proportional to the square root of the restoring force over distance (or the spring constant of a resonator; details can be found in Supplementary Information). In our case, the frequency shift would directly relate to the electric field induced by space charge (E(u) in Equation (10)) divided by distance (u). The absolution value of E(u)/u has a maximum at zero and
(a)
4
Amplitude(mm)
Amplitude(mm)
(b)
4
N = 5 105 N = 1 106 N = 1.5 106
3
503
2
1
qz = 0.36 qz = 0.46 qz = 0.56 3
2
1
0 -30
-25
-20
-15
-10
-5
0
0 -30
-25
-20
f-f0(KHz)
(c)
3
m = 195 m = 350 m = 525
2
1
0 -30
-25
-20
-15
f-f0(KHz)
-10
-5
0
(d)
4
Amplitude(mm)
Amplitude(mm)
4
-15
f-f0(KHz)
-10
-5
0
3
q=1 q=2 q=3
e e e
2
1
0 -30
-25
-20
-15
-10
-5
0
f-f0(KHz)
Figure 3. Nonlinear resonance curves induced by space charge effects in the 3D ion trap. (a) qz = 0.46, m/z =195 Da, q =1×e; (b) n = 1 × 1 06, m/z = 195 Da, q = 1 × e; (c) n = 1 × 106, qz = 0.46, q = 1 × e; (d) n = 1 × 106, qz = 0.46, m/z = 195 Da. T = 300 K, r0 = 5 mm, z0 = 3.536 mm, helium pressure 1 mTorr
504
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
(a)
(b)
Ion cloud size (normalized STD) 0.5
0.2 105
106 N
0.2 0.3 107
12
0.16
N=5 N=1 N=5 N=1
105 106 106 107
normalized STD
104 mm-3) 0.1
0.3
Ion density (
0.05
qx
0.4
(c)
15
9
6
0.12
0.08
3 0.04
0 -1.0
-0.5
0.0
0.5
x/x0
1.0
0
2
4
6
8
Ion number
10 10 6
Figure 4. Ion density distributions in the linear ion trap. (a) Normalized SD of ion clouds with different values of qx and ion abundance; (b) Ion density distributions along the x-axis with different ion abundance; (c) ion cloud size versus ion number. T = 300 K, r0 = 4 mm, m/z = 195 Da
decreases as ion motion amplitude (a1) increases. On the other hand, the value of E/u for a single high-order field could be expressed as un (n = 2 for an octopole field), which will be zero at trap center and increase as ion motion amplitude increases. Radio frequency trapping voltage will affect ion cloud size as well as ion motion frequency. Shown in Figure 2b, higher qz will result in bigger frequency shifts at small ion motion amplitudes and smaller frequency shifts at large ion motion amplitudes. As qz increases, a deeper trapping potential well depth is formed, which leads to a condensed ion cloud (Figure 1a). As a result, ions will experience stronger Coulomb forces with smaller motion amplitudes.
Nonlinear Resonance Curve It is known that the resonance curve of an ion would have an “ocean wave” shape with the presence of high-order fields, which is the nonlinear resonance curve [53]. In this study, it is found for the first time that space charge will also induce the “ocean wave” shape nonlinear resonance curve, but with different characteristics. As shown in Figure 3, the nonlinear resonance curves will shift overall towards the left (the lower frequency side) and bend towards f0 (the secular frequency of an ion when not considering space charge effects). Therefore, when considering space charge effects, an ion will get excited at a frequency lower than f0 during a dipolar excitation/ejection process. With increasing motion amplitudes, its motion frequency will approach f0, which agrees with the phenomena shown in Figure 2. It should be noticed that the mass shift observed in an experiment relates to the frequency shift when ion motion reaches the boundary of an ion trap (“tip” of the resonance curve), which is much smaller than the frequency shifts at low ion motion amplitudes (“root” of
the resonance curve). The nonlinear resonance curve induced by space charge is different from that induced by a high-order field, which centers at f0, bends in both directions for an even-order field, and towards the left for an odd-order field [44, 51]. Figure 3a plots the nonlinear resonance curves for ions with m/z = 195 Da at a qz value of 0.46 as ion number increases. Stronger space charge effects shift the nonlinear resonance curve further left. As ion number increases from 5 × 105 to 1.5 × 106, the “root” of the resonance curve would shift from −11.07 to −16.56 KHz, and from −0.32 to −0.72 KHz for the “tip” of the resonance curve. With the same number of ions, an ion excited at higher qz values have bigger frequency shifts with low motion amplitudes, and smaller frequency shifts as its motion amplitude approaches the trap boundary (Figure 3b). This is also due to the fact that a more condensed ion cloud will be formed at higher qz values, as discussed earlier in Figure 2b. Under the same working conditions, ions with different m/z ratios will have different resonance features. There are two ways for ions to have different m/z ratios: (1) ions with the same number of charge but different masses; (2) ions
Table 1. The Comparison of Ion Frequency Shifts Under Different Ion Abundance Between the 3D and the Linear Ion Traps Frequency shift (KHz) Ion number (n)
3D ion trap
Linear ion trap
1 × 106 1.5 × 106 2 × 106
−2.592 −3.536 −4.355
−0.42 −0.632 −0.843
qu = 0.36, T = 300 K, m/z = 195 Da, q = 1× e. Ion motion amplitude z = z0 = 3.536 mm in the 3D ion trap, and x = x0 = 4 mm in the linear ion trap
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
(a)
4
(b) 4
N=1 N=2 N=3
106 106 106
qx = 0.26 qx = 0.36 qx = 0.46
3
Amplitude(mm)
Amplitude(mm)
3
505
2
1
0 -40
-30
-20
-10
2
1
0 -35
0
-28
f-f0(KHz)
-21
-14
-7
0
f-f0(KHz)
Figure 5. Nonlinear resonance curves induced by space charge effects in the linear ion trap. (a) qx = 0.36, m/z = 195 Da, q = 1 × e; (b) n = 2 × 106, m/z = 195 Da, q = 1 × e. T = 300 K, r0 = 4 mm, helium pressure 1 mTorr
For big biological molecules (peptides and proteins), a molecule would normally have the capability of possessing different number of charges using the electrospray ionization method. When molecules have the same mass but possess different number of charges, inertia of these ions is the same. Keeping other parameters the same (qz = 0.46, N = 106, m = 1050 Da), stronger space charge effects exist for ions possessing more charges, which lead to more frequency shifts as shown in Figure 3d.
with the same mass but different number of charges. In the first case, the resonance curve will be influenced by two factors, the size of the ion cloud and inertia of the ion. From Equation (5), the pseudopotential well depth is deeper for heavier ions at the same qz value. Similar to the case in Figure 2b, deeper potential well depth leads to denser ion cloud and, thus, stronger space charge effects on an ion at low motion amplitudes. Furthermore, heavier ions will have larger inertia, and will be influenced less significantly by the same force (Equation (20)). Figure 3c plots the nonlinear resonance curves for three types of singly charged ions with mass 195, 350, and 525, respectively. Under the calculation condition (qz = 0.46, N = 1 × 106), the frequency shift of light ions is larger than that of heavy ions, which indicates that the inertia of an ion will dominate.
4
qz = 0.36, T = 300K qz = 0.46, T = 2000K qz = 0.56, T = 4000K
Amplitude(mm)
Amplitude(mm)
Ion Cloud Size An ion cloud inside a linear ion trap has a cylindrical shape and Gaussian distributions in the x- and y-directions of the ion trap, a uniform distribution in the z-direction of the ion trap. The trapping volume of the linear
(a)
4
3
Linear Ion Trap
2
3
2
1
1
0 -25
(b) qx = 0.36, T = 300K qx = 0.46, T = 2000K qx = 0.56, T = 4000K
-20
-15
-10
f-f0(KHz)
-5
0
0 -35
-28
-21
-14
-7
0
f-f0(KHz)
Figure 6. Minimization of space charge induced frequency shifts. (a) The nonlinear resonance curves in the 3D ion trap with increased qz and T, N, = 106; (b) the nonlinear resonance curves in the linear ion trap with increased qx and T, N, = 2 × 106. m/z = 195 Da, q = 1 × e, helium pressure 1 mTorr
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D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
ion trap is bigger than that of the 3D ion trapl; more ions could be trapped inside the linear ion trap. Similar to the situation of the 3D ion trap, the size of an ion cloud gets bigger as the value of qx decreases or the number of ions increases as, shown in Figure 4a. When occupying the same relative volume of an ion trap (the normalized SD of the ion cloud), the linear ion trap can trap about an order of magnitude more ions than the 3D ion trap. With an increasing number of ions, ion cloud will expand, as well as the ion density (Figure 4b and c).
Ion Secular Frequency Shift In the calculation, a cloud of ion with m/z = 195 Da was first placed in the linear ion trap, then the motion of a single ion (m/z = 195 Da) was studied at different motional amplitudes. The electric field distribution induced by space charge along the x- or y-axis of the linear ion trap is similar to that within the 3D ion trap (Supplementary Figure S1), so space charge has similar effects in the linear ion trap. As shown in Supplementary Figure S2, ion motion frequency will decrease when considering space charge effects, and bigger frequency shifts with more ions. With the presence of 2 × 106 ions in the linear ion trap, Supplementary Figure S2b plots the frequency shifts of the ion at different qx values. Similar to the 3D ion trap cases shown in Figure 2b, lower motion amplitudes lead to bigger frequency shifts. Different from Figure 2b, ion motion frequency shifts more at low qx values with small motion amplitudes as shown in Supplementary Figure S2b. This is due to the fact that higher q values result in both stronger quadrupole field (deeper potential well depth, Du) and denser ion clouds at the center of the ion trap. Denser ion clouds will cause stronger space charge effects. The electric field attributable to space charge has a negative quadrupole field component (Equation (16)). Therefore, it is a counterbalance between these two effects as q increases. In those cases, shown in Supplementary Figure S2b, stronger quadrupole field effects dominate the ion motion frequency shifts at low motion amplitudes. When the same number of ions are trapped, the frequency shift are larger in the 3D ion trap than those in the linear ion trap as shown in Table 1, which corresponds to the larger trapping capacity of the linear ion trap.
Nonlinear Resonance Curve Similar nonlinear resonance curves are also present for ions excited in the linear ion trap. Figure 5a and b show the effects of q value and ion number on the nonlinear resonance curves. As ion number increases from 1 × 106 to 3 × 106 (qx = 0.36), the “root” of the resonance curves would shift from −14.1 to −35.73 KHz, and from −0.32 to −0.72 KHz at the “tip” of the resonance curves. As the value of qx increases from 0.26 to 0.46 (N = 2 × 106), the “root” of the resonance curves would shift from −28.6 to −22.74 KHz, and from −1.69 to −0.96 KHz at the “tip” of the resonance curves. When trapped the same number of ions, ion frequency shift is bigger in the 3D ion trap than that in the linear ion trap.
Minimization of Space Charge Effects During a typical ion trap operation, it would be beneficial to minimize space charge induced effects. As discussed earlier, ion ejection or operation at higher q values would typically reduce ion motion frequency shifts, especially at higher ion motion amplitudes (Figure 2b, 3b, Supplementary S2b, and Figure 5b). Another factor that would reduce space charge effects is the temperature of the ion cloud. Based on Equations (13) and (14), an ion cloud would have a bigger radius at higher temperatures, which results in weaker Coulomb interactions within the ion cloud. As an example, Figure 6a and b show that the frequency shifts originating from space charge effects could be reduced in both 3D and linear ion traps with increased q value and ion cloud temperature, which agrees with the findings from experiments [33]. Mathematically, mass resolution could be defined as m/△m or ω/△ω. An approximate expression of mass resolution (ω/△ω) can be derived based on Equation (19) (Equation S30 in Supplementary Information), which shows the dependence of mass resolution on q values and ion temperature. With reduced space charge effects, better mass resolutions are expected. As an indicator of peak broadening effect, the widths of the resonance curves get narrower with increased q values and ion temperature. It should be noticed that the ion temperature may not equal to the physical temperature of the ion trap. Under thermo equilibrium, ion temperature is equal to (or closest to) the physical temperature of the ion trap without considering the rf heating effects [54, 55]. However, with excitations, velocity of an ion could be high, and its equivalent temperature could be much higher than that of the surrounding environment. In order to separate space charge effects from higher-order field effects, ideal ion traps with pure quadrupole electric field were considered in this work. However, higher-order field will exist in a practical ion trap because of electrode truncation, ejection slits, and surface roughness, etc. [2, 56–58]. As shown earlier in Equation (16), the electric field induced by space charge is a summation of even-order field components. Although space charge would enhance or cancel the effects introduced by high-order fields, it may not be practical to utilize higher-order fields to minimize space charge effects or vice versa. The high-order field components attributable to ion trap geometry imperfection have fixed values (or fixed percentages). On the other hand, the electric field induced by space charge has a dynamic characteristic, since the ion cloud distribution and ion number will change with the varying working conditions (rf voltage, for example) during a typical ion trap operation process.
Conclusion For the first time, it is shown theoretically that space charge will induce nonlinear effects on ion trajectories within 3D and linear ion traps. A group of ions has a close to Gaussian
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
distribution in the quadrupole electric field. The electric field generated by space charge can be expressed as a summation of even-order fields, which results in the “ocean wave” shape non-linear resonance curves for ions under dipolar excitations. As a net effect, space charge will always lower ion motion frequency. Larger frequency shifts exist at low motion amplitudes, smaller frequency shifts at high motion amplitudes, which is just the opposite effect caused by any single high-order field. Higher q values (from the Mathieu equation) and temperatures (ion cloud temperature) could be utilized to reduce space charge effects.
20.
Acknowledgments
21.
This work was supported by National Natural Sciences Foundation of China (21205005) and National Scientific Instrumentation Grant Program of China (2011YQ09000502, 2011YQ09000501 and 2012YQ040140-07).
16. 17. 18.
19.
22. 23. 24.
References 1. Cooks, R.G.: Special feature: historical collision-induced dissociation: readings and commentary. J. Mass Spectrom. 30(9), 1215–1221 (1995) 2. March, R.E.: Quadrupole Ion Trap Mass Spectrometer. Wiley Online Library: (2000) 3. Coon, J.J., Shabanowitz, J., Hunt, D.F., Syka, J.E.P.: Electron transfer dissociation of peptide anions. J. Am. Soc. Mass Spectrom. 16(6), 880– 882 (2005) 4. Huzarska, M., Ugalde, I., Kaplan, D.A., Hartmer, R., Easterling, M.L., Polfer, N.C.: Negative electron transfer dissociation of deprotonated phosphopeptide anions: choice of radical cation reagent and competition between electron and proton transfer. Anal. Chem. 82(7), 2873–2878 (2010) 5. Vincent, C.E., Rensvold, J.W., Westphall, M.S., Pagliarini, D.J., Coon, J.J.: Automated gas-phase purification for accurate, multiplexed quantification on a stand-alone ion-trap mass spectrometer. Anal. Chem. 85(4), 2079–2086 (2013) 6. Vedel, F., Andre, J.: Influence of space charge on the computed statistical properties of stored ions cooled by a buffer gas in a quadrupole rf trap. Phys. Rev. A 29(4), 2098 (1984) 7. Cox, K.A., Cleven, C.D., Cooks, R.G.: Mass shifts and local space charge effects observed in the quadrupole ion trap at higher resolution. Int. J. Mass Spectrom. Ion Processes 144(1/2), 47–65 (1995) 8. Xiong, X.C., Xu, W., Fang, X., Deng, Y.L., Ouyang, Z.: Accelerated Simulation study of space charge effects in quadrupole ion traps using GPU techniques. J. Am. Soc. Mass Spectrom. 23(10), 1799–1807 (2012) 9. Douglas, D.J., Frank, A.J., Mao, D.M.: Linear ion traps in mass spectrometry. Mass Spectrom. Rev. 24(1), 1–29 (2005) 10. Plass, W.R., Li, H.Y., Cooks, R.G.: Theory, simulation, and measurement of chemical mass shifts in rf quadrupole ion traps. Int. J. Mass Spectrom. 228(2/3), 237–267 (2003) 11. Schwartz, J.C., Senko, M.W., Syka, J.E.P.: A two-dimensional quadrupole ion trap mass spectrometer. J. Am. Soc. Mass Spectrom. 13(6), 659–669 (2002) 12. Vedel, F., Andre, J., Vedel, M., Brincourt, G.: Computed energy and spatial statistical properties of stored ions cooled by a buffer gas. Phys. Rev. A 27(5), 2321 (1983) 13. Guan, S., Marshall, A.: Equilibrium space charge distribution in a quadrupole ion trap. J. Am. Soc. Mass Spectrom. 5(2), 64–71 (1994) 14. Hemberger, P.H., Nogar, N.S., Williams, J.D., Cooks, R.G., Syka, J.E.P.: Laser photodissociation probe for ion tomography studies in a quadrupole ion-trap mass spectrometer. Chem. Phys. Lett. 191(5), 405– 410 (1992) 15. Cleven, C.D., Cooks, R.G., Garrett, A.W., Nogar, N.S., Hemberger, P.H.: Radial distributions and ejection times of molecular ions in an ion
25. 26.
27.
28.
29. 30. 31. 32.
33. 34.
35.
36. 37.
38.
507
trap mass spectrometer: a laser tomography study of effects of ion density and molecular type. J. Phys. Chem. 100(1), 40–46 (1996) Plass, W.R., Gill, L.A., Bui, H.A., Cooks, R.G.: Ion mobility measurement by DC tomography in an rf quadrupole ion trap. J. Phys. Chem. A 104(21), 5059–5065 (2000) Parks, J.H., Szoke, A.: Simulation of collisional relaxation of trapped ion clouds in the presence of space charge fields. J. Chem. Phys. 103(4), 1422–1439 (1995) Grinfeld, D.E., Kopaev, I.A., Makarov, A.A., Monastyrskiy, M.A.: Equilibrium ion distribution modeling in rf ion traps and guides with regard to Coulomb effects. Nuclear Instrum. Methods Phys. Res. A 645(1), 141–145 (2011) Hager, J.W.: A new linear ion trap mass spectrometer. Rapid Commun. Mass Spectrom. 16(6), 512–526 (2002) Nikolaev, E.N., Heeren, R.M.A., Popov, A.M., Pozdneev, A.V., Chingin, K.S.: Realistic modeling of ion cloud motion in a Fourier transform ion cyclotron resonance cell by use of a particle-in-cell approach. Rapid Commun. Mass Spectrom. 21(22), 3527 (2007) Ledford, E.B., Rempel, D.L., Gross, M.L.: Space charge effects in Fourier transform mass spectrometry. II. Mass calibration. Anal. Chem. 56(14), 2744–2748 (1984) Amster, I.J.: Fourier transform mass spectrometry. J. Mass Spectrom. 31(12), 1325–1337 (1996) Comisarow, M.B., Marshall, A.G.: Frequency-sweep Fourier transform ion cyclotron resonance spectroscopy. Chem. Phys. Lett. 26(4), 489– 490 (1974) Comisarow, M.B., Marshall, A.G.: Fourier transform ion cyclotron resonance spectroscopy. Chem. Phys. Lett. 25(2), 282–283 (1974) Jeffries, J.B., Barlow, S.E., Dunn, G.H.: Theory of space-charge shift of ion cyclotron resonance frequencies. Int. J. Mass Spectrom. Ion Processes 54(1/2), 169–187 (1983) Nikolaev, E.N., Miluchihin, N.V., Inoue, M.: Evolution of an ion cloud in a Fourier transform ion cyclotron resonance mass spectrometer during signal detection: its influence on spectral line shape and position. Int. J. Mass Spectrom. Ion Processes 148(3), 145–157 (1995) Kharchenko, A., Vladimirov, G., Heeren, R.M.A., Nikolaev, E.N.: Performance of Orbitrap mass analyzer at various space charge and nonideal field conditions: simulation approach. J. Am. Soc. Mass Spectrom. 23(5), 977–987 (2012) Makarov, A., Denisov, E., Kholomeev, A., Baischun, W., Lange, O., Strupat, K., Horning, S.: Performance evaluation of a hybrid linear ion trap/Orbitrap mass spectrometer. Anal. Chem. 78(7), 2113–2120 (2006) Hu, Q.Z., Noll, R.J., Li, H.Y., Makarov, A., Hardman, M., Cooks, R.G.: The Orbitrap: a new mass spectrometer. J. Mass Spectrom. 40(4), 430– 443 (2005) Ding, L., Sudakov, M., Kumashiro, S.: A simulation study of the digital ion trap mass spectrometer. Int. J. Mass Spectrom. 221(2), 117–138 (2002) Ding, L., Sudakov, M., Brancia, F.L., Giles, R., Kumashiro, S.: A digital ion trap mass spectrometer coupled with atmospheric pressure ion sources. J. Mass Spectrom. 39(5), 471–484 (2004) McLuckey, S.A., Wu, J., Bundy, J.L., Stephenson, J.L., Hurst, G.B.: Oligonucleotide mixture, analysis via electrospray and ion/ion reactions in a quadrupole ion trap. Anal. Chem. 74(5), 976–984 (2002) Qiao, H., Gao, C., Mao, D., Konenkov, N., Douglas, D.J.: Space-charge effects with mass-selective axial ejection from a linear quadrupole ion trap. Rapid Commun. Mass Spectrom. 25(23), 3509–3520 (2011) Todd, J.F.J., Waldren, R.M., Freer, D.A., Turner, R.B.: The quadrupole ion store (QUISTOR). Part X. Space charge and ion stability. B. On the theoretical distribution and density of stored charge in rf quadrupole fields. Int. J. Mass Spectrom. Ion Phys. 35(1), 107–150 (1980) Todd, J.F.J., Waldren, R.M., Mather, R.E.: The quadrupole ion store (QUISTOR). Part IX. Space charge and ion stability. A. Theoretical background and experimental results. Int. J. Mass Spectrom. Ion Physics 34(3), 325–349 (1980) Vedel, F.: On the dynamics and energy of ion clouds stored in an rf quadrupole trap. Int. J. Mass Spectrom. Ion Processes 106, 33–61 (1991) Li, G.Z., Guan, S.H., Marshall, A.G.: Comparison of equilibrium ion density distribution and trapping force in Penning, Paul, and combined ion traps. J. Am. Soc. Mass Spectrom. 9(5), 473–481 (1998) Tolmachev, A.V., Udseth, H.R., Smith, R.D.: Charge capacity limitations of radio frequency ion guides in their use for improved ion accumulation and trapping in mass spectrometry. Anal. Chem. 72(5), 970–978 (2000)
508
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
39. Prestage, J.D., Dick, G.J., Maleki, L.: New ion trap for frequency standard applications. J. Appl. Phys. 66(3), 1013–1017 (1989) 40. Campbell, J.M., Collings, B.A., Douglas, D.J.: A new linear ion trap time-of-flight system with tandem mass spectrometry capabilities. Rapid Commun. Mass Spectrom. 12(20), 1463–1474 (1998) 41. Chen, S.-P., Comisarow, M.B.: Simple physical models for coulombinduced frequency shifts and coulomb-induced inhomogeneous broadening for like and unlike ions in Fourier transform ion cyclotron resonance mass spectrometry. Rapid Commun. Mass Spectrom. 5(10), 450–455 (1991) 42. Chen, S.-P., Comisarow, M.B.: Modeling Coulomb effects in Fourier-transform ion cyclotron resonance mass spectrometry by charged disks and charged cylinders. Rapid Commun. Mass Spectrom. 6(1), 1–3 (1992) 43. Mitchell, D.W., Smith, R.D.: Cyclotron motion of two Coulombically interacting ion clouds with implications to Fourier transform ion cyclotron resonance mass spectrometry. Phys. Rev. E. 52(4), 4366 (1995) 44. Wang, Y., Huang, Z., Jiang, Y., Xiong, X., Deng, Y., Fang, X., Xu, W.: The coupling effects of hexapole and octopole fields in quadrupole ion traps: a theoretical study. J. Mass Spectrom. 48(8), 937–944 (2013) 45. Mickens, R.E.: A generalization of the method of harmonic balance. J. Sound Vibration 111(3), 515–518 (1986) 46. Dehmelt, H.G.: Radiofrequency spectroscopy of stored ions. I: Storage. Adv. At. Mol. Phys. 3, 53 (1967) 47. Stafford Jr., G.C., Kelley, P.E., Syka, J.E.P., Reynolds, W.E., Todd, J.F.J.: Recent improvements in and analytical applications of advanced ion trap technology. Int. J. Mass Spectrom. Ion Processes 60(1), 85–98 (1984) 48. Xu, W., Chappell, W.J., Ouyang, Z.: Modeling of ion transient response to dipolar AC excitation in a quadrupole ion trap. Int. J. Mass Spectrom. 308(1), 49–55 (2011)
49. Doroudi, A., Asl, A.R.: Calculation of secular axial frequencies in a nonlinear ion trap with hexapole, octopole, and decapole superpositions by a modified Lindstedt-Poincare method. Int. J. Mass Spectrom. 309, 104–108 (2012) 50. Zhou, X., Zhu, Z., Xiong, C., Chen, R., Xu, W., Qiao, H., Peng, W.-P., Nie, Z., Chen, Y.: Characteristics of stability boundary and frequency in nonlinear ion trap mass spectrometer. J. Am. Soc. Mass Spectrom. 21(9), 1588–1595 (2010) 51. Zhao, X., Douglas, D.J.: Dipole excitation of ions in linear radio frequency quadrupole ion traps with added multipole fields. Int. J. Mass Spectrom. 275(1/3), 91–103 (2008) 52. Sevugarajan, S., Menon, A.G.: Frequency perturbation in nonlinear Paul traps: a simulation study of the effect of geometric aberration, space charge, dipolar excitation, and damping on ion axial secular frequency. Int. J. Mass Spectrom. 197(1), 263–278 (2000) 53. Makarov, A.A.: Resonance ejection from the Paul trap: a theoretical treatment incorporating a weak octapole field. Anal. Chem. 68(23), 4257–4263 (1996) 54. Goeringer, D.E., Viehland, L.A., Danailov, D.M.: Prediction of collective characteristics for ion ensembles in quadrupole ion traps without trajectory simulations. J. Am. Soc. Mass Spectrom. 17(7), 889–902 (2006) 55. Prentice, B.M., McLuckey, S.A.: Dipolar DC collisional activation in a "stretched" 3D ion trap: the effect of higher order fields on rf heating. J. Am. Soc. Mass Spectrom. 23(4), 736–744. 56. Lammert, S.A., Plass, W.R., Thompson, C.V., Wise, M.B.: Design, optimization, and initial performance of a toroidal rf ion trap mass spectrometer. Int. J. Mass Spectrom. 212(1/3), 25–40 (2001) 57. Beaty, E.C.: Calculated electrostatic properties of ion traps. Phys. Rev. A 33(6), 3645 (1986) 58. Xu, W., Chappell, W.J., Cooks, G.R., Ouyang, Z.: Characterization of electrode surface roughness and its impact on ion trap mass analysis. J. Mass Spectrom. 44, 353–360 (2009)
J. Am. Soc. Mass Spectrom. (2014) 25:498Y508 DOI: 10.1007/s13361-013-0784-9
RESEARCH ARTICLE
Space Charge Induced Nonlinear Effects in Quadrupole Ion Traps Dan Guo,1 Yuzhuo Wang,1 Xingchuang Xiong,2 Hua Zhang,3 Xiaohua Zhang,3 Tao Yuan,4 Xiang Fang,2 Wei Xu1,5 1
Department of Biomedical Engineering, Beijing Institute of Technology, Beijing, China National Institute of Metrology, Beijing, China 3 Beijing Purkinje General Instrument Co., Ltd, Beijing, China 4 KunShan Innowave Communication Technology Co., Ltd, Jiangshu, China 5 School of Life Science, Beijing Institute of Technology, Haidian, Beijing, 100081, China 2
Abstract. A theoretical method was proposed in this work to study space charge effects in quadrupole ion traps, including ion trapping, ion motion frequency shift, and nonlinear effects on ion trajectories. The spatial distributions of ion clouds within quadrupole ion traps were first modeled for both 3D and linear ion traps. It is found that the electric field generated by space charge can be expressed as a summation of even-order fields, such as quadrupole field, octopole field, etc. Ion trajectories were then solved using the harmonic balance method. Similar to highorder field effects, space charge will result in an “ocean wave” shape nonlinear resonance curve for an ion under a dipolar excitation. However, the nonlinear resonance curve will be totally shifted to lower frequencies and bend towards ion secular frequency as ion motion amplitude increases, which is just the opposite effect of any even-order field. Based on theoretical derivations, methods to reduce space charge effects were proposed. Key words: Space charge, Quadrupole ion trap, Nonlinear resonance curve, Harmonic balance method Received: 27 August 2013/Revised: 28 October 2013/Accepted: 9 November 2013/Published online: 3 January 2014
Introduction
C
ombined with collision induced dissociation (CID) [1, 2] and electron transfer dissociation (ETD) [3–5] techniques, quadrupole ion traps have been widely used in mass spectrometry (MS) systems as mass analyzers and/or ion trapping reaction devices. As an ion manipulation device, the performance of a quadrupole ion trap is greatly influenced by space charge effects. With increasing number of ions trapped inside a quadrupole ion trap, Coulomb force between ions will modify ion trajectory and result in chemical mass shift, resolution degradation, and etc. [6–11]. Great efforts have been made to explore and minimize space charge effects in quadrupole ion traps.
Dan Guo and Yuzhuo Wang contributed equally to this work. Electronic supplementary material The online version of this article (doi:10.1007/s13361-013-0784-9) contains supplementary material, which is available to authorized users. Correspondence to: Wei Xu; e-mail: [email protected]
To understand space charge effects, simulation and experimental researches have been carried out. Due to buffer gas cooling effect, an ion cloud will reach thermal equilibrium inside a quadrupole ion trap [12, 13]. Laser tomography experiments [14–16] and simulations [8, 17, 18] have been performed to study the spatial distributions of ions within 3D quadrupole ion traps, which showed that ion clouds have close to Gaussian distributions along the radial and axial directions under equilibrium. Simulation and experiments have also shown that space charge will cause chemical mass shift, limited dynamic range, peak covalence, and resolution degradation not only in quadrupole ion traps [7–11, 15, 16, 19] but also in Fourier transform ion cyclotron resonance (FT-ICR) cells [20–26], Orbitraps [27–29], and digital ion traps [30, 31]. On the other hand, Coulomb force between ions possessing opposite polarities has also been utilized to enhance ion trapping [32]. Experiments were also performed to investigate the best operating conditions in linear ion traps to minimize space charge induced mass shift and resolution degradation [11, 33]. Besides simulation and experiments, theoretical modeling would be a powerful tool to gain an in-depth understanding of space charge effects [6, 34–36].
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
Utilizing the classic Boltzmann distribution, ion spatial distributions within quadrupole and Penning ion traps have been calculated under thermal equilibrium conditions [13, 17, 18, 33, 37]. Ion trapping capacity of quadrupole ion guides (when used as ion trapping devices) and quadrupole ion traps were studied through balancing the effective rf trapping and space charge repulsion fields [9, 13, 37, 38]. Based on a relatively straightforward trapping volume calculation, the ratios of the maximum number of ions stored in a linear ion trap to that in a 3D ion trap were estimated [11, 39, 40]. The interactions between two ion clouds in FT-ICR cells have been explored by modeling an ion cloud with point charge, line charge, uniform charged disk, uniform charged cylinder, and uniform charged sphere, respectively [41–43]. However, because of the complexity of Coulomb interactions (for instance, ion motion equation will become nonlinear when including Coulomb forces), it still lacks effective methods of studying space charge effects on ion motion in quadrupole ion traps. In this work, theoretical models have been first proposed to study the ion cloud distribution within both 3D and linear ion traps. Furthermore, space charge effects on ion motion characteristics have been explored, including ion motional frequency shift and the nonlinear resonance curve. Starting from Boltzmann distribution, an ion cloud is found to have close to Gaussian distributions along the quadrupole field directions within both 3D and linear ion traps. In this study, the electrical field originated from space charge effects was expressed as a summation of high-order fields, and the harmonic balance method [44, 45] was then used to solve the ion motion equation. Space charge-induced ion frequency shift and nonlinear resonance curve were explored. With the knowledge obtained from space charge modeling, it is found that ion motion frequency shifts originated from Coulomb interaction could be reduced by operating at higher q values and higher ion cloud temperatures.
Experimental Ion Cloud Modeling To characterize space charge effects on ion motions in quadrupole ion traps, ion distributions within both 3D and linear ion traps were first investigated. When trapped inside a quadrupole ion trap, ions will form an ion cloud. With the pseudopotential approximation, the ion density distribution (or ion spatial distribution) in an ion trap meets Boltzmann distribution under thermal equilibrium [13], which can be written as:
nðx; y; zÞ ¼ N
∫
−q ϕ ðx; y; zÞ exp KT −q ϕ ðx; y; zÞ dxdydz exp KT
ð1Þ
499
where N is the total number of ions, q is the electric charge that an ion possesses, K is the Boltzmann constant, T is the temperature of the ion cloud and ϕ represents the electric potential within the ion trap. Since the denominator is a constant, Equation (1) can be simplified as: nðx; y; zÞ ¼ N 0 exp
−qϕ ðx; y; zÞ KT
ð2Þ
The electric potential ϕ can be written as the summation of rf (radio frequency) trapping potential (ϕI) and the potential induced by space charge (ϕS). ϕ ¼ ϕI þ ϕS
ð3Þ
To isolate space charge effects from high-order field effects, ion traps with pure quadrupole electric field were considered in this work. According to the pseudopotential approximation theory, the rf trapping potential in a 3D ion trap can be expressed as [46]: ϕI ¼
Dr 2 Dz 2 r þ 2z þC r20 z0
ð4Þ
in which r0 is the radius of the ring electrode, z0 is the distance from trap center to end cap and C is a constant. Dr and Dz represent the pseudopotential well depth along the radial (r) and axial (z) directions, and Dr ¼
mr20 Ω 2 q2 mz2 Ω 2 q2 a x þ x ; Dz ¼ 0 az þ z ð5Þ 2q 2 2 2q 2 2
where ax, az, and qx, qz are the dimensionless constants in the Mathieu Equation, Ω is the rf frequency. Since Dr/r02 may not equal to Dz/z02, the ion cloud within a 3D ion trap has an elliptical shape in most cases. If Dr/r02 = Dz/z02 or at a higher q values, an ion cloud could form a spherical shape [8]. For simplicity, a spherical ion cloud shape is assumed within the 3D ion trap in this study, and the rf trapping potential can be approximated as [13],
ϕI ¼
Dr 2 u þC r20
ð6Þ
where u is the distance from the point of interest to trap center. In a linear ion trap, the rf trapping potential in the x-y plane can be expressed as,
ϕI ¼
Dr 2 r þC r20
ð7Þ
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D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
where au is set to zero. Along the z-axis, a zero electric potential is assumed within the effective trapping length of a linear ion trap, and a uniform ion distribution is applied. With controlled number of ions, the potential induced by space charge is small comparing to the rf trapping potential, and the ion cloud distributions could be approximated as Gaussian distributions [13]. Substituting Equations (6) and (7) back to Equation (2), Equation (2) reduces to the form of a Gaussian distribution for ions with the same m/z (mass-tocharge) ratio. Therefore, in a 3D ion trap, ion density distribution could be expressed as, 2 N −u nðuÞ ¼ pffiffiffiffiffi 3 exp 2a2 2π a
ð8Þ
where a is the standard deviation (SD). In a linear ion trap, ion density distribution in the x-y plane could be written as,
nð r Þ ¼
2 ρl −r exp 2a2 π 2a2
ð9Þ
in which ρl is the ion line density, ρl = N/z0 and z0 is the effective length of the linear ion trap. With the ion distributions shown in Equations (8) and (9), the electric field (E) induced by space charge could be calculated using Gaussian theorem (details in Supplementary Information): 8
u > > u2 p ffiffi ffi qN Erf > − > > e 2a2 2a < E ðuÞ ¼ − pqN ffiffiffi 3D ion trap þ 4ε0 πu2 2 2aε0 π 3=2 u > 2
> > qρl −r > > 1 − exp linear ion trap :E ðrÞ ¼ 2πrε0 2a2 ð10Þ where Erf is the error function and ε0 is the permittivity of vacuum. The corresponding electric potential induced by space charge, ϕS can then be obtained by integrating the electric field (details in Supplementary Information). 8
u > > p ffiffi ffi NqErf > > > 2a
Z > ∞ > qρl −r2 > > dr linear ion trap 1− exp : ϕs ¼ 2πrε0 2a2 r ð11Þ
To obtain an exact ion distribution within an ion trap, an iterative process needs to be taken: solving the electric potential caused by the ion density distribution; substituting the electric potential into Equation (2) and a new ion density
distribution is computed; and repeating the above process numerically until two successive iteration results agree with each other. Both experimental measurements [14–16] and simulations [8, 17, 18] have shown that ion clouds have close to Gaussian distributions within a quadrupole ion trap. As a first order approximation, the above iteration was carried out once in this work to get an analytical solution as well as simplifying the calculation process. Substituting Equation (6), (7), (11) into Equation (2) and applying the second order Taylor expansion on the electric potential induced by space charge (ϕS), SD of the ion cloud distributions could be calculated, 8 1 −q Dr Q > > > 3D ion trap < − 2a2 ¼ KT r2 − pffiffiffi 3 3=2 12 2a π ε0 0 > 1 −q Dr ρl q > > − linear ion trap :− 2 ¼ 2a KT r20 8 a2 πε0 ð12Þ in which Q is the total charge inside the 3D ion trap, Q = qN. Thus rffiffiffi 3 2 pffiffiffi k π T ε0 ρ20 3 a¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=3 pffiffiffi 3 2Dr 2 q3 Qε20 ρ20 þ 18Dr 4 q6 Q2 ε40 ρ40 − 96Dr 3 k 3 π 3 q3 T 3 ε60 ρ60 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3 2Dr 2 q3 Qε20 ρ20 þ 18Dr 4 q6 Q2 ε40 ρ40 − 96Dr 3 k 3 π 3 q3 T 3 ε60 ρ60 1=3 p ffiffiffiffiffiffiffiffi pffiffiffi þ 3 144Dr π qε0
ð13Þ in a 3D ion trap, and ρ a¼ 0 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4kπT ε0 þ q2 ρl 2πqε0 Dr
ð14Þ
in a linear ion trap.
Ion Motion Modeling After knowing the ion cloud distribution and space charge induced electric field, space charge effects on ion motion could be studied. In this work, ion motion along the ejection axis was explored, for instance the z-axis in a 3D ion trap, the x or y-axis in a linear ion trap. Applying the pseudopotential approximation and considering space charge Coulomb force, ion motion inside an ion trap can be written as: u00 þ ω20 u ¼
q E m
ð15Þ
where u could be z in a 3D ion trap and x or y in a linear ion trap; ω0 is the secular frequency of the ion (angular
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
frequency); E is the electric field induced by space charge as shown in Equation (10). In order to solve Equation (15), the electric field E was expanded using an 11th order polynomial fitting method. E ðuÞ ¼ p1 u þ p3 u3 þ p5 u5 þ p7 u7 þ p9 u9 þ p11 u11
ð16Þ
where pn (n = 1, 3, 5, 7, 9, 11) are polynomial coefficients. The electric field is an odd function (electric potential an even function), since the Coulomb force between either positive or negative ions is always pushing ions away from the center. Space charge-induced electric field can be well represented by the 11th order polynomial fitting in both 3D and linear ion traps (details in Supplementary Information). Equation (16) indicates that the electric field induced by space charge can be treated as a summation of even-order fields. The corresponding relationship between high-order field components and the polynomial coefficients in Equation (16) can be found in Supplementary Information. Replacing E in Equation (15) with Equation (16), u00 þ ω20 u ¼ P1 u þ P3 u3 þ P5 u5 þ P7 u7 þ P9 u9 þ P11 u11
ð17Þ
where P ¼ mq p , Equation (17) can be solved directly by applying the harmonic balance method, which was introduced earlier in exploring high-order field effects in ion traps [44, 45]. Briefly, with the first-order approximation, the solution of Equation (17) can be written as u ¼ a0 þ a1 cosðωt Þ
ð18Þ
where a0 and a1 are coefficients of the zero-order and firstorder harmonic functions; ω is the ion motion angular frequency. Ion motion characteristics and trajectory can be calculated by substituting Equation (18) back into Equation (17) and balancing the coefficients of harmonic functions (details in Supplementary Information). With the presence of space charge Coulomb force, ion motion angular frequency will be modified as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3P3 a21 5P5 a41 35P7 a61 63P9 a81 231P11 a10 1 ω ¼ ω20 − P1 − − − − − 4 8 64 128 512
ð19Þ
During a typical ion ejection and detection process, AC (alternating current) dipolar excitation and buffer gas damping effects need to be considered. Modeling ion-
501
molecule collisions with a damping term (c), ion motion equation becomes 0 u00 þ cu þ ω20 u ¼ P1 u þ P3 u3 þ P5 u5 þ P7 u7 þ P9 u9 þ P11 u11 þ Pac cosðωac t Þ
ð20Þ
in which ωac is the angular frequency of the AC excitation signal; Pac accounts for the excitation strength, which was well defined in earlier works [47, 48]. Details about the calculations of Pac can be found in Supplementary Information). Following a similar method, the solution of Equation (20) can be written as u ¼ a0 þ a1 cosðωac t Þ þ a2 sinðωac t Þ
ð21Þ
The relationship among AC excitation frequency, ion motion frequency and amplitude (the nonlinear resonance curve) can be obtained after solving Equation (20) using the harmonic balance method (details in Supplementary Information). The 3D ion trap studied in this work has the dimensions of r0 = 5 mm, z0 = 3.535 mm; while the linear ion trap has the dimensions of x0 = y0 = 4 mm and z0 = 30 mm. The rf signals applied on both ion traps have a frequency of 1 MHz, and the rf voltage will be adjusted accordingly as qu was changed. Helium was used as the buffer gas with a pressure of 1 mTorr. Calculations were based on ions with m/z = 195 Da at a temperature of 300 K, unless otherwise specified.
Measurement of Errors In the derivation process, the pseudopotential approximation and the assumption that an ion cloud has a close to Gaussian distribution were used. The pseudopotential approximation has been widely used and tested to be accurate at low q values [46]. It has also been verified by theoretical [13], numerical [8, 17, 18], and experimental studies [14–16] that an ion cloud will have a close to Gaussian distribution with controlled number of ions. In this work, both of the above conditions were tried to be maintained in later calculations. Besides the two assumptions above, there are two major approximations: (1) the use of polynomial fitting to approximate the electric field induced by space charge; (2) the use of the harmonic balance method to solve the ion motion equation. The accuracy of this theoretical method is estimated by comparing the calculated frequency shifts with those obtained from a numerical method (details in Supplementary Information). Results show that the calculation error is within ~2 % for the 3D ion trap and ~6 % for the linear ion trap under typical operation conditions.
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D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
(b)
(a)
qz
0.1 0.4
0.2
0.32
104 105 105 106
N=5 N=1 N=5 N=1
normalized STD
104 mm-3)
0.05
0.5
(c)
9
Ion density (
0.6
Ion cloud size (normalized STD)
6
3
0.3 103
104
105
106
0.16
0.08
0.3 0.4
0.24
0 -1.0
107
-0.5
0.0
0.5
1.0
0
1
z/z0
N
2
3
Ion number
4
5 106
Figure 1. Ion density distributions in the 3D ion trap. (a) Normalized SD of ion clouds with different values of qz and ion abundance; (b) ion density distributions along the z-axis under different ion abundance; (c) Ion cloud size versus ion number. T = 300 K, r0 = 5 mm, z0 = 3.536 mm, m/z = 195 Da. As qz changes from 0.3 to 0.6, the rf voltage was swept from ~149 to 299 V0-p
cloud can be characterized using the SD (a). Calculated from Equation (13), Figure 1a plots the ion cloud size (normalized SD) versus the total ion number (N) and rf trapping condition (qz) in the 3D ion trap. Under the pseudopotential approximation, the ion trap has deeper trapping potential well depth at higher qz values, which results in tighter ion clouds. Figure 1b plots the ion distributions inside the ion trap at the same qz value but different ion numbers. As ion number increases, the ion cloud would expand rapidly with low ion numbers and slower with high ion numbers (Figure 1c), which agrees with simulation [8] and experimental results [15].
Results and Discussion Previous simulation and experimental results have shown that space charge would cause mass shift, resolution degradation and peak coalescence [7–11, 15, 16, 19]. Expressed as a summation of high-order fields as shown in Equation (16), space charge would also introduce nonlinear phenomena on ion motion characteristics.
3D Ion Trap Ion Cloud Size When cooled inside an ion trap, the size of an ion cloud has strong dependence on the rf trapping field and number of ions. While the Coulomb force pushes ions away from each other, the rf trapping field would compress ions towards the center of an ion trap. With a Gaussian distribution, the size of an ion
Ion Secular Frequency The effects of space charge on ion motion frequency were characterized by solving Equa-
(a)
0
N = 5 105 N = 1 10 6 N = 1.5 106
-5
qz = 0.36 qz = 0.46 qz = 0.56
-7
f-f0(KHz)
f-f0(KHz)
(b)
0
-10
-15
-14
-20
-21 0.0
0.2
0.4
z/z0
0.6
0.8
1.0
-25 0.0
0.2
0.4
0.6
0.8
1.0
z/z0
Figure 2. Ion secular frequency shifts with different ion abundance and qz = 0.46 (a); at different qz values and n = 1 × 106 (b). T = 300 K, r0 = 5 mm, z0 = 3.536 mm, m/z = 195 Da, q = 1 × e
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
tion (19) at different ion motion amplitudes (z), ion numbers (N), and qz values. In the calculation, a group of ions with m/ z = 195 Da was first placed in the 3D ion trap, then the motion of a single ion (m/z = 195 Da) was studied at different motional amplitudes. In general, ion motion frequency will decrease with the presence of Coulomb forces as shown in Figure 2, which agrees with simulation and experimental observations [7, 8, 10]. The space charge induced electric field (in Equation (10)) can be expressed in a polynomial form (Equation (16)), which can be seen as a summation of even-order fields (such as quadrupole, octopole, etc.). Furthermore, the Coulomb force will always push ions away from trap center, which resembles a net negative even-order field. Therefore, space charge effects will weaken the quadrupole field (P1 term in Equation (10)), as well as adding higher-order fields into the ion trap (P3, P5 … terms in Equation (10)). Mathematically, P1, P5, P9 … in Equation (16) are positive and P3, P7, P11 … are negative. In Equations (16) and (19), P1, P5, and P7 terms will be larger than P3, P7, and P11 terms, respectively. Thus, with weakened quadrupole field dominating, the modified ion motion frequency is always lower than the ion motion frequency without considering space charge effects.
While sharing similar features with high-order field effects, space charge effects are different from those arisen from any single high-order field component. Figure 2a plots the ion motion frequency shifts versus ion motion amplitudes at different ion numbers. The relative frequency shift from the ideal value (ion secular frequency without considering space charge effects f0) is larger at lower ion motion amplitudes and smaller at higher motion amplitudes, which are just the opposite effects caused by any even-order field [49, 50]. For instance, with an added octopole field, ion motion frequency will center at its secular frequency (f0) at low ion motion amplitudes, and shifts away from the secular frequency as ion motion amplitude increases [44, 51, 52]. A potential explanation is that with the pseudopotential approximation, ion motion within an ion trap can be treated as a harmonic oscillator. The characteristic frequency of a harmonic oscillator is proportional to the square root of the restoring force over distance (or the spring constant of a resonator; details can be found in Supplementary Information). In our case, the frequency shift would directly relate to the electric field induced by space charge (E(u) in Equation (10)) divided by distance (u). The absolution value of E(u)/u has a maximum at zero and
(a)
4
Amplitude(mm)
Amplitude(mm)
(b)
4
N = 5 105 N = 1 106 N = 1.5 106
3
503
2
1
qz = 0.36 qz = 0.46 qz = 0.56 3
2
1
0 -30
-25
-20
-15
-10
-5
0
0 -30
-25
-20
f-f0(KHz)
(c)
3
m = 195 m = 350 m = 525
2
1
0 -30
-25
-20
-15
f-f0(KHz)
-10
-5
0
(d)
4
Amplitude(mm)
Amplitude(mm)
4
-15
f-f0(KHz)
-10
-5
0
3
q=1 q=2 q=3
e e e
2
1
0 -30
-25
-20
-15
-10
-5
0
f-f0(KHz)
Figure 3. Nonlinear resonance curves induced by space charge effects in the 3D ion trap. (a) qz = 0.46, m/z =195 Da, q =1×e; (b) n = 1 × 1 06, m/z = 195 Da, q = 1 × e; (c) n = 1 × 106, qz = 0.46, q = 1 × e; (d) n = 1 × 106, qz = 0.46, m/z = 195 Da. T = 300 K, r0 = 5 mm, z0 = 3.536 mm, helium pressure 1 mTorr
504
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
(a)
(b)
Ion cloud size (normalized STD) 0.5
0.2 105
106 N
0.2 0.3 107
12
0.16
N=5 N=1 N=5 N=1
105 106 106 107
normalized STD
104 mm-3) 0.1
0.3
Ion density (
0.05
qx
0.4
(c)
15
9
6
0.12
0.08
3 0.04
0 -1.0
-0.5
0.0
0.5
x/x0
1.0
0
2
4
6
8
Ion number
10 10 6
Figure 4. Ion density distributions in the linear ion trap. (a) Normalized SD of ion clouds with different values of qx and ion abundance; (b) Ion density distributions along the x-axis with different ion abundance; (c) ion cloud size versus ion number. T = 300 K, r0 = 4 mm, m/z = 195 Da
decreases as ion motion amplitude (a1) increases. On the other hand, the value of E/u for a single high-order field could be expressed as un (n = 2 for an octopole field), which will be zero at trap center and increase as ion motion amplitude increases. Radio frequency trapping voltage will affect ion cloud size as well as ion motion frequency. Shown in Figure 2b, higher qz will result in bigger frequency shifts at small ion motion amplitudes and smaller frequency shifts at large ion motion amplitudes. As qz increases, a deeper trapping potential well depth is formed, which leads to a condensed ion cloud (Figure 1a). As a result, ions will experience stronger Coulomb forces with smaller motion amplitudes.
Nonlinear Resonance Curve It is known that the resonance curve of an ion would have an “ocean wave” shape with the presence of high-order fields, which is the nonlinear resonance curve [53]. In this study, it is found for the first time that space charge will also induce the “ocean wave” shape nonlinear resonance curve, but with different characteristics. As shown in Figure 3, the nonlinear resonance curves will shift overall towards the left (the lower frequency side) and bend towards f0 (the secular frequency of an ion when not considering space charge effects). Therefore, when considering space charge effects, an ion will get excited at a frequency lower than f0 during a dipolar excitation/ejection process. With increasing motion amplitudes, its motion frequency will approach f0, which agrees with the phenomena shown in Figure 2. It should be noticed that the mass shift observed in an experiment relates to the frequency shift when ion motion reaches the boundary of an ion trap (“tip” of the resonance curve), which is much smaller than the frequency shifts at low ion motion amplitudes (“root” of
the resonance curve). The nonlinear resonance curve induced by space charge is different from that induced by a high-order field, which centers at f0, bends in both directions for an even-order field, and towards the left for an odd-order field [44, 51]. Figure 3a plots the nonlinear resonance curves for ions with m/z = 195 Da at a qz value of 0.46 as ion number increases. Stronger space charge effects shift the nonlinear resonance curve further left. As ion number increases from 5 × 105 to 1.5 × 106, the “root” of the resonance curve would shift from −11.07 to −16.56 KHz, and from −0.32 to −0.72 KHz for the “tip” of the resonance curve. With the same number of ions, an ion excited at higher qz values have bigger frequency shifts with low motion amplitudes, and smaller frequency shifts as its motion amplitude approaches the trap boundary (Figure 3b). This is also due to the fact that a more condensed ion cloud will be formed at higher qz values, as discussed earlier in Figure 2b. Under the same working conditions, ions with different m/z ratios will have different resonance features. There are two ways for ions to have different m/z ratios: (1) ions with the same number of charge but different masses; (2) ions
Table 1. The Comparison of Ion Frequency Shifts Under Different Ion Abundance Between the 3D and the Linear Ion Traps Frequency shift (KHz) Ion number (n)
3D ion trap
Linear ion trap
1 × 106 1.5 × 106 2 × 106
−2.592 −3.536 −4.355
−0.42 −0.632 −0.843
qu = 0.36, T = 300 K, m/z = 195 Da, q = 1× e. Ion motion amplitude z = z0 = 3.536 mm in the 3D ion trap, and x = x0 = 4 mm in the linear ion trap
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
(a)
4
(b) 4
N=1 N=2 N=3
106 106 106
qx = 0.26 qx = 0.36 qx = 0.46
3
Amplitude(mm)
Amplitude(mm)
3
505
2
1
0 -40
-30
-20
-10
2
1
0 -35
0
-28
f-f0(KHz)
-21
-14
-7
0
f-f0(KHz)
Figure 5. Nonlinear resonance curves induced by space charge effects in the linear ion trap. (a) qx = 0.36, m/z = 195 Da, q = 1 × e; (b) n = 2 × 106, m/z = 195 Da, q = 1 × e. T = 300 K, r0 = 4 mm, helium pressure 1 mTorr
For big biological molecules (peptides and proteins), a molecule would normally have the capability of possessing different number of charges using the electrospray ionization method. When molecules have the same mass but possess different number of charges, inertia of these ions is the same. Keeping other parameters the same (qz = 0.46, N = 106, m = 1050 Da), stronger space charge effects exist for ions possessing more charges, which lead to more frequency shifts as shown in Figure 3d.
with the same mass but different number of charges. In the first case, the resonance curve will be influenced by two factors, the size of the ion cloud and inertia of the ion. From Equation (5), the pseudopotential well depth is deeper for heavier ions at the same qz value. Similar to the case in Figure 2b, deeper potential well depth leads to denser ion cloud and, thus, stronger space charge effects on an ion at low motion amplitudes. Furthermore, heavier ions will have larger inertia, and will be influenced less significantly by the same force (Equation (20)). Figure 3c plots the nonlinear resonance curves for three types of singly charged ions with mass 195, 350, and 525, respectively. Under the calculation condition (qz = 0.46, N = 1 × 106), the frequency shift of light ions is larger than that of heavy ions, which indicates that the inertia of an ion will dominate.
4
qz = 0.36, T = 300K qz = 0.46, T = 2000K qz = 0.56, T = 4000K
Amplitude(mm)
Amplitude(mm)
Ion Cloud Size An ion cloud inside a linear ion trap has a cylindrical shape and Gaussian distributions in the x- and y-directions of the ion trap, a uniform distribution in the z-direction of the ion trap. The trapping volume of the linear
(a)
4
3
Linear Ion Trap
2
3
2
1
1
0 -25
(b) qx = 0.36, T = 300K qx = 0.46, T = 2000K qx = 0.56, T = 4000K
-20
-15
-10
f-f0(KHz)
-5
0
0 -35
-28
-21
-14
-7
0
f-f0(KHz)
Figure 6. Minimization of space charge induced frequency shifts. (a) The nonlinear resonance curves in the 3D ion trap with increased qz and T, N, = 106; (b) the nonlinear resonance curves in the linear ion trap with increased qx and T, N, = 2 × 106. m/z = 195 Da, q = 1 × e, helium pressure 1 mTorr
506
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
ion trap is bigger than that of the 3D ion trapl; more ions could be trapped inside the linear ion trap. Similar to the situation of the 3D ion trap, the size of an ion cloud gets bigger as the value of qx decreases or the number of ions increases as, shown in Figure 4a. When occupying the same relative volume of an ion trap (the normalized SD of the ion cloud), the linear ion trap can trap about an order of magnitude more ions than the 3D ion trap. With an increasing number of ions, ion cloud will expand, as well as the ion density (Figure 4b and c).
Ion Secular Frequency Shift In the calculation, a cloud of ion with m/z = 195 Da was first placed in the linear ion trap, then the motion of a single ion (m/z = 195 Da) was studied at different motional amplitudes. The electric field distribution induced by space charge along the x- or y-axis of the linear ion trap is similar to that within the 3D ion trap (Supplementary Figure S1), so space charge has similar effects in the linear ion trap. As shown in Supplementary Figure S2, ion motion frequency will decrease when considering space charge effects, and bigger frequency shifts with more ions. With the presence of 2 × 106 ions in the linear ion trap, Supplementary Figure S2b plots the frequency shifts of the ion at different qx values. Similar to the 3D ion trap cases shown in Figure 2b, lower motion amplitudes lead to bigger frequency shifts. Different from Figure 2b, ion motion frequency shifts more at low qx values with small motion amplitudes as shown in Supplementary Figure S2b. This is due to the fact that higher q values result in both stronger quadrupole field (deeper potential well depth, Du) and denser ion clouds at the center of the ion trap. Denser ion clouds will cause stronger space charge effects. The electric field attributable to space charge has a negative quadrupole field component (Equation (16)). Therefore, it is a counterbalance between these two effects as q increases. In those cases, shown in Supplementary Figure S2b, stronger quadrupole field effects dominate the ion motion frequency shifts at low motion amplitudes. When the same number of ions are trapped, the frequency shift are larger in the 3D ion trap than those in the linear ion trap as shown in Table 1, which corresponds to the larger trapping capacity of the linear ion trap.
Nonlinear Resonance Curve Similar nonlinear resonance curves are also present for ions excited in the linear ion trap. Figure 5a and b show the effects of q value and ion number on the nonlinear resonance curves. As ion number increases from 1 × 106 to 3 × 106 (qx = 0.36), the “root” of the resonance curves would shift from −14.1 to −35.73 KHz, and from −0.32 to −0.72 KHz at the “tip” of the resonance curves. As the value of qx increases from 0.26 to 0.46 (N = 2 × 106), the “root” of the resonance curves would shift from −28.6 to −22.74 KHz, and from −1.69 to −0.96 KHz at the “tip” of the resonance curves. When trapped the same number of ions, ion frequency shift is bigger in the 3D ion trap than that in the linear ion trap.
Minimization of Space Charge Effects During a typical ion trap operation, it would be beneficial to minimize space charge induced effects. As discussed earlier, ion ejection or operation at higher q values would typically reduce ion motion frequency shifts, especially at higher ion motion amplitudes (Figure 2b, 3b, Supplementary S2b, and Figure 5b). Another factor that would reduce space charge effects is the temperature of the ion cloud. Based on Equations (13) and (14), an ion cloud would have a bigger radius at higher temperatures, which results in weaker Coulomb interactions within the ion cloud. As an example, Figure 6a and b show that the frequency shifts originating from space charge effects could be reduced in both 3D and linear ion traps with increased q value and ion cloud temperature, which agrees with the findings from experiments [33]. Mathematically, mass resolution could be defined as m/△m or ω/△ω. An approximate expression of mass resolution (ω/△ω) can be derived based on Equation (19) (Equation S30 in Supplementary Information), which shows the dependence of mass resolution on q values and ion temperature. With reduced space charge effects, better mass resolutions are expected. As an indicator of peak broadening effect, the widths of the resonance curves get narrower with increased q values and ion temperature. It should be noticed that the ion temperature may not equal to the physical temperature of the ion trap. Under thermo equilibrium, ion temperature is equal to (or closest to) the physical temperature of the ion trap without considering the rf heating effects [54, 55]. However, with excitations, velocity of an ion could be high, and its equivalent temperature could be much higher than that of the surrounding environment. In order to separate space charge effects from higher-order field effects, ideal ion traps with pure quadrupole electric field were considered in this work. However, higher-order field will exist in a practical ion trap because of electrode truncation, ejection slits, and surface roughness, etc. [2, 56–58]. As shown earlier in Equation (16), the electric field induced by space charge is a summation of even-order field components. Although space charge would enhance or cancel the effects introduced by high-order fields, it may not be practical to utilize higher-order fields to minimize space charge effects or vice versa. The high-order field components attributable to ion trap geometry imperfection have fixed values (or fixed percentages). On the other hand, the electric field induced by space charge has a dynamic characteristic, since the ion cloud distribution and ion number will change with the varying working conditions (rf voltage, for example) during a typical ion trap operation process.
Conclusion For the first time, it is shown theoretically that space charge will induce nonlinear effects on ion trajectories within 3D and linear ion traps. A group of ions has a close to Gaussian
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
distribution in the quadrupole electric field. The electric field generated by space charge can be expressed as a summation of even-order fields, which results in the “ocean wave” shape non-linear resonance curves for ions under dipolar excitations. As a net effect, space charge will always lower ion motion frequency. Larger frequency shifts exist at low motion amplitudes, smaller frequency shifts at high motion amplitudes, which is just the opposite effect caused by any single high-order field. Higher q values (from the Mathieu equation) and temperatures (ion cloud temperature) could be utilized to reduce space charge effects.
20.
Acknowledgments
21.
This work was supported by National Natural Sciences Foundation of China (21205005) and National Scientific Instrumentation Grant Program of China (2011YQ09000502, 2011YQ09000501 and 2012YQ040140-07).
16. 17. 18.
19.
22. 23. 24.
References 1. Cooks, R.G.: Special feature: historical collision-induced dissociation: readings and commentary. J. Mass Spectrom. 30(9), 1215–1221 (1995) 2. March, R.E.: Quadrupole Ion Trap Mass Spectrometer. Wiley Online Library: (2000) 3. Coon, J.J., Shabanowitz, J., Hunt, D.F., Syka, J.E.P.: Electron transfer dissociation of peptide anions. J. Am. Soc. Mass Spectrom. 16(6), 880– 882 (2005) 4. Huzarska, M., Ugalde, I., Kaplan, D.A., Hartmer, R., Easterling, M.L., Polfer, N.C.: Negative electron transfer dissociation of deprotonated phosphopeptide anions: choice of radical cation reagent and competition between electron and proton transfer. Anal. Chem. 82(7), 2873–2878 (2010) 5. Vincent, C.E., Rensvold, J.W., Westphall, M.S., Pagliarini, D.J., Coon, J.J.: Automated gas-phase purification for accurate, multiplexed quantification on a stand-alone ion-trap mass spectrometer. Anal. Chem. 85(4), 2079–2086 (2013) 6. Vedel, F., Andre, J.: Influence of space charge on the computed statistical properties of stored ions cooled by a buffer gas in a quadrupole rf trap. Phys. Rev. A 29(4), 2098 (1984) 7. Cox, K.A., Cleven, C.D., Cooks, R.G.: Mass shifts and local space charge effects observed in the quadrupole ion trap at higher resolution. Int. J. Mass Spectrom. Ion Processes 144(1/2), 47–65 (1995) 8. Xiong, X.C., Xu, W., Fang, X., Deng, Y.L., Ouyang, Z.: Accelerated Simulation study of space charge effects in quadrupole ion traps using GPU techniques. J. Am. Soc. Mass Spectrom. 23(10), 1799–1807 (2012) 9. Douglas, D.J., Frank, A.J., Mao, D.M.: Linear ion traps in mass spectrometry. Mass Spectrom. Rev. 24(1), 1–29 (2005) 10. Plass, W.R., Li, H.Y., Cooks, R.G.: Theory, simulation, and measurement of chemical mass shifts in rf quadrupole ion traps. Int. J. Mass Spectrom. 228(2/3), 237–267 (2003) 11. Schwartz, J.C., Senko, M.W., Syka, J.E.P.: A two-dimensional quadrupole ion trap mass spectrometer. J. Am. Soc. Mass Spectrom. 13(6), 659–669 (2002) 12. Vedel, F., Andre, J., Vedel, M., Brincourt, G.: Computed energy and spatial statistical properties of stored ions cooled by a buffer gas. Phys. Rev. A 27(5), 2321 (1983) 13. Guan, S., Marshall, A.: Equilibrium space charge distribution in a quadrupole ion trap. J. Am. Soc. Mass Spectrom. 5(2), 64–71 (1994) 14. Hemberger, P.H., Nogar, N.S., Williams, J.D., Cooks, R.G., Syka, J.E.P.: Laser photodissociation probe for ion tomography studies in a quadrupole ion-trap mass spectrometer. Chem. Phys. Lett. 191(5), 405– 410 (1992) 15. Cleven, C.D., Cooks, R.G., Garrett, A.W., Nogar, N.S., Hemberger, P.H.: Radial distributions and ejection times of molecular ions in an ion
25. 26.
27.
28.
29. 30. 31. 32.
33. 34.
35.
36. 37.
38.
507
trap mass spectrometer: a laser tomography study of effects of ion density and molecular type. J. Phys. Chem. 100(1), 40–46 (1996) Plass, W.R., Gill, L.A., Bui, H.A., Cooks, R.G.: Ion mobility measurement by DC tomography in an rf quadrupole ion trap. J. Phys. Chem. A 104(21), 5059–5065 (2000) Parks, J.H., Szoke, A.: Simulation of collisional relaxation of trapped ion clouds in the presence of space charge fields. J. Chem. Phys. 103(4), 1422–1439 (1995) Grinfeld, D.E., Kopaev, I.A., Makarov, A.A., Monastyrskiy, M.A.: Equilibrium ion distribution modeling in rf ion traps and guides with regard to Coulomb effects. Nuclear Instrum. Methods Phys. Res. A 645(1), 141–145 (2011) Hager, J.W.: A new linear ion trap mass spectrometer. Rapid Commun. Mass Spectrom. 16(6), 512–526 (2002) Nikolaev, E.N., Heeren, R.M.A., Popov, A.M., Pozdneev, A.V., Chingin, K.S.: Realistic modeling of ion cloud motion in a Fourier transform ion cyclotron resonance cell by use of a particle-in-cell approach. Rapid Commun. Mass Spectrom. 21(22), 3527 (2007) Ledford, E.B., Rempel, D.L., Gross, M.L.: Space charge effects in Fourier transform mass spectrometry. II. Mass calibration. Anal. Chem. 56(14), 2744–2748 (1984) Amster, I.J.: Fourier transform mass spectrometry. J. Mass Spectrom. 31(12), 1325–1337 (1996) Comisarow, M.B., Marshall, A.G.: Frequency-sweep Fourier transform ion cyclotron resonance spectroscopy. Chem. Phys. Lett. 26(4), 489– 490 (1974) Comisarow, M.B., Marshall, A.G.: Fourier transform ion cyclotron resonance spectroscopy. Chem. Phys. Lett. 25(2), 282–283 (1974) Jeffries, J.B., Barlow, S.E., Dunn, G.H.: Theory of space-charge shift of ion cyclotron resonance frequencies. Int. J. Mass Spectrom. Ion Processes 54(1/2), 169–187 (1983) Nikolaev, E.N., Miluchihin, N.V., Inoue, M.: Evolution of an ion cloud in a Fourier transform ion cyclotron resonance mass spectrometer during signal detection: its influence on spectral line shape and position. Int. J. Mass Spectrom. Ion Processes 148(3), 145–157 (1995) Kharchenko, A., Vladimirov, G., Heeren, R.M.A., Nikolaev, E.N.: Performance of Orbitrap mass analyzer at various space charge and nonideal field conditions: simulation approach. J. Am. Soc. Mass Spectrom. 23(5), 977–987 (2012) Makarov, A., Denisov, E., Kholomeev, A., Baischun, W., Lange, O., Strupat, K., Horning, S.: Performance evaluation of a hybrid linear ion trap/Orbitrap mass spectrometer. Anal. Chem. 78(7), 2113–2120 (2006) Hu, Q.Z., Noll, R.J., Li, H.Y., Makarov, A., Hardman, M., Cooks, R.G.: The Orbitrap: a new mass spectrometer. J. Mass Spectrom. 40(4), 430– 443 (2005) Ding, L., Sudakov, M., Kumashiro, S.: A simulation study of the digital ion trap mass spectrometer. Int. J. Mass Spectrom. 221(2), 117–138 (2002) Ding, L., Sudakov, M., Brancia, F.L., Giles, R., Kumashiro, S.: A digital ion trap mass spectrometer coupled with atmospheric pressure ion sources. J. Mass Spectrom. 39(5), 471–484 (2004) McLuckey, S.A., Wu, J., Bundy, J.L., Stephenson, J.L., Hurst, G.B.: Oligonucleotide mixture, analysis via electrospray and ion/ion reactions in a quadrupole ion trap. Anal. Chem. 74(5), 976–984 (2002) Qiao, H., Gao, C., Mao, D., Konenkov, N., Douglas, D.J.: Space-charge effects with mass-selective axial ejection from a linear quadrupole ion trap. Rapid Commun. Mass Spectrom. 25(23), 3509–3520 (2011) Todd, J.F.J., Waldren, R.M., Freer, D.A., Turner, R.B.: The quadrupole ion store (QUISTOR). Part X. Space charge and ion stability. B. On the theoretical distribution and density of stored charge in rf quadrupole fields. Int. J. Mass Spectrom. Ion Phys. 35(1), 107–150 (1980) Todd, J.F.J., Waldren, R.M., Mather, R.E.: The quadrupole ion store (QUISTOR). Part IX. Space charge and ion stability. A. Theoretical background and experimental results. Int. J. Mass Spectrom. Ion Physics 34(3), 325–349 (1980) Vedel, F.: On the dynamics and energy of ion clouds stored in an rf quadrupole trap. Int. J. Mass Spectrom. Ion Processes 106, 33–61 (1991) Li, G.Z., Guan, S.H., Marshall, A.G.: Comparison of equilibrium ion density distribution and trapping force in Penning, Paul, and combined ion traps. J. Am. Soc. Mass Spectrom. 9(5), 473–481 (1998) Tolmachev, A.V., Udseth, H.R., Smith, R.D.: Charge capacity limitations of radio frequency ion guides in their use for improved ion accumulation and trapping in mass spectrometry. Anal. Chem. 72(5), 970–978 (2000)
508
D. Guo et al.: Space Charge Modeling in Quadrupole Ion Traps
39. Prestage, J.D., Dick, G.J., Maleki, L.: New ion trap for frequency standard applications. J. Appl. Phys. 66(3), 1013–1017 (1989) 40. Campbell, J.M., Collings, B.A., Douglas, D.J.: A new linear ion trap time-of-flight system with tandem mass spectrometry capabilities. Rapid Commun. Mass Spectrom. 12(20), 1463–1474 (1998) 41. Chen, S.-P., Comisarow, M.B.: Simple physical models for coulombinduced frequency shifts and coulomb-induced inhomogeneous broadening for like and unlike ions in Fourier transform ion cyclotron resonance mass spectrometry. Rapid Commun. Mass Spectrom. 5(10), 450–455 (1991) 42. Chen, S.-P., Comisarow, M.B.: Modeling Coulomb effects in Fourier-transform ion cyclotron resonance mass spectrometry by charged disks and charged cylinders. Rapid Commun. Mass Spectrom. 6(1), 1–3 (1992) 43. Mitchell, D.W., Smith, R.D.: Cyclotron motion of two Coulombically interacting ion clouds with implications to Fourier transform ion cyclotron resonance mass spectrometry. Phys. Rev. E. 52(4), 4366 (1995) 44. Wang, Y., Huang, Z., Jiang, Y., Xiong, X., Deng, Y., Fang, X., Xu, W.: The coupling effects of hexapole and octopole fields in quadrupole ion traps: a theoretical study. J. Mass Spectrom. 48(8), 937–944 (2013) 45. Mickens, R.E.: A generalization of the method of harmonic balance. J. Sound Vibration 111(3), 515–518 (1986) 46. Dehmelt, H.G.: Radiofrequency spectroscopy of stored ions. I: Storage. Adv. At. Mol. Phys. 3, 53 (1967) 47. Stafford Jr., G.C., Kelley, P.E., Syka, J.E.P., Reynolds, W.E., Todd, J.F.J.: Recent improvements in and analytical applications of advanced ion trap technology. Int. J. Mass Spectrom. Ion Processes 60(1), 85–98 (1984) 48. Xu, W., Chappell, W.J., Ouyang, Z.: Modeling of ion transient response to dipolar AC excitation in a quadrupole ion trap. Int. J. Mass Spectrom. 308(1), 49–55 (2011)
49. Doroudi, A., Asl, A.R.: Calculation of secular axial frequencies in a nonlinear ion trap with hexapole, octopole, and decapole superpositions by a modified Lindstedt-Poincare method. Int. J. Mass Spectrom. 309, 104–108 (2012) 50. Zhou, X., Zhu, Z., Xiong, C., Chen, R., Xu, W., Qiao, H., Peng, W.-P., Nie, Z., Chen, Y.: Characteristics of stability boundary and frequency in nonlinear ion trap mass spectrometer. J. Am. Soc. Mass Spectrom. 21(9), 1588–1595 (2010) 51. Zhao, X., Douglas, D.J.: Dipole excitation of ions in linear radio frequency quadrupole ion traps with added multipole fields. Int. J. Mass Spectrom. 275(1/3), 91–103 (2008) 52. Sevugarajan, S., Menon, A.G.: Frequency perturbation in nonlinear Paul traps: a simulation study of the effect of geometric aberration, space charge, dipolar excitation, and damping on ion axial secular frequency. Int. J. Mass Spectrom. 197(1), 263–278 (2000) 53. Makarov, A.A.: Resonance ejection from the Paul trap: a theoretical treatment incorporating a weak octapole field. Anal. Chem. 68(23), 4257–4263 (1996) 54. Goeringer, D.E., Viehland, L.A., Danailov, D.M.: Prediction of collective characteristics for ion ensembles in quadrupole ion traps without trajectory simulations. J. Am. Soc. Mass Spectrom. 17(7), 889–902 (2006) 55. Prentice, B.M., McLuckey, S.A.: Dipolar DC collisional activation in a "stretched" 3D ion trap: the effect of higher order fields on rf heating. J. Am. Soc. Mass Spectrom. 23(4), 736–744. 56. Lammert, S.A., Plass, W.R., Thompson, C.V., Wise, M.B.: Design, optimization, and initial performance of a toroidal rf ion trap mass spectrometer. Int. J. Mass Spectrom. 212(1/3), 25–40 (2001) 57. Beaty, E.C.: Calculated electrostatic properties of ion traps. Phys. Rev. A 33(6), 3645 (1986) 58. Xu, W., Chappell, W.J., Cooks, G.R., Ouyang, Z.: Characterization of electrode surface roughness and its impact on ion trap mass analysis. J. Mass Spectrom. 44, 353–360 (2009)
