The fundamental equations of motion for the ions

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1 Characteristics of Stability Boundary and Frequency in Nonlinear Ion Trap Mass Spectrometer Xiaoyu Zhou, a,b Zhiqiang Zhu, a Caiqiao Xiong, a Rui Chen, a Wenjun Xu, a Haoxue Qiao, b Wen-Ping Peng, c Zongxiu Nie, a and Yi Chen a a Key Laboratory of Analytical Chemistry for Living Biosystems, Institute of Chemistry Chinese Academy of Sciences, Beijing National Laboratory for Molecular Sciences, Beijing, China b Department of Physics, Wuhan University, Wuhan, China c Department of Physics, National Dong Hwa University, Hualien, Taiwan In this article, the Poincare-Lighthill-Kuo (PLK) method is used to derive an analytical expression on the stability boundary and the ion trajectory. A multipole superposition model mainly including octopole component is adopted to represent the inhomogeneities of the field. In this method, both the motional displacement and secular frequency of ions have been expanded to asymptotic series by the scale of nonlinear term, which represents a weak octopole field. By solving the zero and first-order approximate equations, it is found that a frequency shift exists between the ideal and nonlinear conditions. The motional frequency of ions in nonlinear ion trap depends on not only Mathieu parameters, a and q, but also the percentage of the nonlinear field and the initial amplitude of ions. In the same trap, ions have the same mass-to-charge ratio (m/z) but they have different initial amplitudes or velocities. Consequently, they will be ejected at different time through after a mass-selective instability scan. The influences on the mass resolution in quadrupole ion trap, which is coupled with positive or negative octopole fields, have been discussed respectively. (J Am Soc Mass Spectrom 010, 1, ) 010 American Society for Mass Spectrometry The fundamental equations of motion for the ions within the ideal quadrupole potential are exactly characterized by the Mathieu equation [1], the equation gives: u u where u is the related motional frequency in the axial and radial direction, is the frequency of the driven radio-frequency (rf) potential, u is a parameter dependent on Mathieu parameters a u and q u, and the subscript u refers to axial and radial coordinates. Boundary ejection methods take the advantage of ejecting ions with a mass-selective instability scan, with which the rf amplitude is scanned linearly to cause the secular frequency of ions to increase until they become unstable [ 4], when the ions reach the boundary at a 0, q eject 0.908, z is equal to one, the ion has reached its stability limit. When a buffer gas was first used to improve the resolution of the mass spectrum by Stafford et al. [4], the stability boundary would move towards the positive direction on q-axis as bath gas pressure increase [, 5, 6]. For the particle under the background at about 40 mtorr, the delay ejection also occurred at a large q value Address reprint requests to Dr. Z. X. Nie and Dr. Y. Chen, Institute of Chemistry Chinese Academy of Sciences, Beijing , China. znie@iccas.ac.cn and chenyi@iccas.ac.cn. in the audio-frequency ion trap mass spectrometer [7 11]. In practical traps, however, the electric field distribution is nonlinear, and the main causes contributing to field distortions are the misalignment of the trap [1], the truncation of electrodes [1], and the space charge [13, 14]. As the existence of high order terms in electric field, the stability boundary deviates from the ideal value and the equation of motion for the ions will be nonlinear Mathieu equation. Most papers [15 19] deal with the time-dependent nonlinear terms with the method of pseudopotential well approximation [0]. eff e m dt. Using this approximation, high order terms become time-independent, and the nonlinear Mathieu equation is a normal nonlinear equation known as Duffing equation. The solution of such kind of equations has been well-studied [1]. When q u 0.4 [, ], this approximation accords well with the physical truth, some useful results such as ion trajectories, oscillation frequency, and the relation between the mass resolution and scan frequency by the technique of resonance ejection have been obtained [3 5]. However, this Published online April 4, American Society for Mass Spectrometry. Published by Elsevier Inc. Received February 4, /10/$3.00 Revised April 1, 010 doi: /j.jasms Accepted April 1, 010

2 J Am Soc Mass Spectrom 010, 1, CHARACTERISTICS IN NONLINEAR ION TRAP MS 1589 approximation is not suitable for discussing the situation of high q value and stability region boundary []. Over the past decades, it has been demonstrated that the quadrupole trap is an excellent micro-laboratory for investigation of nonlinear dynamics, both experimentally and theoretically [6 8]. Quadrupole field coupled with octopole and other multipoles fields have been widely adopted as theory model for studying mass resolution in a practical ion trap [16, 18, 3]. Douglas and coworkers have studied the high order multipoles effects in the trapping potentials by changing the geometry and the distance of the trap electrodes. The results have shown that the mass resolution improves dramatically with added octopole field in proper experimental conditions [9 3]. Such characterizations were successfully investigated by adding octopole fields to the main quadrupole field [33], and theoretical study has demonstrated the positive octopole component will accelerate ion ejection improving the mass resolution [34]. The purpose of this article is to study the stability boundary and the frequency characters in the nonlinear quadrupole trap by directly solving nonlinear Mathieu equation using Poincare-Lighthill-Kuo (PLK) method [35, 36]. The nonlinear effects mainly induced by octopole component are also discussed. The perturbation of the secular frequency and the trajectories of the ions have been investigated by the perturbation methods as the nonlinear term is far smaller than the linear term. As the influence of the nonlinear term, the ions with the same mass-to-charge ratio (m/z) but different initial amplitudes will not be ejected synchronously. It leads to the spitting of the stability boundary and influencing the mass resolution. An expression has been proposed to discuss the relative factors to the stability boundary. Theory In the practical operating mode, the trapped field U-Vcost is input between the end-caps and the ring electrodes, where V stands for the zero to peak of amplitude voltage with frequency, U denotes the dc voltage. The motion of the ions in the practical quadrupole ion trap can be described by a nonlinear Mathieu equation [17]: d u d (a u q u cos )(u u 3 ) 0 (1) where the dimensionless Mathieu parameters a, q, are defined as a z a r 8eU mr 0 q z q r 4eV mr 0 t and the nonlinear parameter is defined as A 4 A r 0 where A and A 4 are the dimensionless amplitudes of quadrupole potential and octopole potential, respectively [15, 18], r 0 is the radius of the ring electrode, e is ion charge and m is mass of ion. By the method of pseudopotential well approximation, the eq 1 becomes the Duffing equation [1]. Generally, the nonlinear oscillation has different time scale because of the nonlinear restoring force. Owing to the interaction between different frequency oscillations caused by nonlinear effect, the amplitude varies slowly with time while the phase varies fast. The perturbed oscillation frequency depends on not only the initial secular frequency but the initial amplitude. So the frequency-dependent variables of ion motion and u displacement-dependent variables of ion motion can be expanded to an asymptotic series in terms of. After understanding the physical meaning of the nonlinear oscillation, the same perturbation procedure is chosen to solve nonlinear Mathieu equation. Define variable (see Aappendix A), and eq 1 becomes d u d (a q cos )(u u3 ) 0 () Using PLK method and choosing as perturbation parameter, eq can be solved and the variables u and are both expanded to asymptotic series in terms of : u u 0 u 1 (3) 0 1 (4) When the nonlinear term turns to zero or 0, an ideal model is obtained with exactly the parameters of amplitude-dependent u 0 and frequency-dependent 0. Differently, since the existence of nonlinear terms, eq 1 produces the amplitude and frequency shift, which deviate from its ideal value u 0 and 0, respectively, and the perturbed amplitude and the frequency should be expanded in terms of. Substituting eq 3 and eq 4 into eq and comparing the order of, the zero, first-order approximation equations are 0 d u 0 d (a q cos )u 0 0 (5)

3 1590 ZHOU ET AL. J Am Soc Mass Spectrom 010, 1, d u 1 0 d d u d (a q cos )(u 1 u 3 0 ) 0 (6) Define a a 0, q q (7) 0 The zero-order approximation equation becomes d u 1 d (a q cos )u 1 (a q cos ) C C 0 The non-duration condition of eq 10 is So cos C 0 3 cos 3 (10) C 0 0 (11) d u 0 d (a q cos )u 0 0 (8) This is a linear Mathieu equation with the new parameters a= and q=, which deviate from the normal parameters a and q due to the octopole superposition, and eq 7 shows their dependency on the deviation of frequency. The solutions of the unperturbed equation are known as Mathieu functions [1, ]: u 0 () C n cos(n ) S n sin(n ) (9) where and are arbitrary constants, C n and S n are recursively defined constant coefficients, and = is related to the Mathieu parameters a= and q=. From eqs 8 and 9, the motion behavior of ions in a practical nonlinear trap is much similar to that in an ideal linear trap with a perturbed secular frequency. Through solving eq 6 in combination with eq, the perturbed frequency could be calculated. In eq 9, since the solution is a polynomial of sine and cosine function, with the coefficient of the latter term is far smaller than that of the previous C 0 C C 4...,thecontribution of the high order terms can be neglected. When the initial velocity of ions is small enough, it satisfies S n 0. As the amplitudes of macro- and micro-motion in the firstorder approximation is a small value for the perturbation comparing to zero-order approximation, such treatment in first-order approximation is reasonable for simplifying the calculation, and the results would be accurate in high q value. Especially, this treatment could accurately reflect the motional characters of the ions. Substituting eq 5 and eq 9 into eq 6, we have: C 0 0 (1) where C 0 is the initial amplitude, so the perturbed a= and q= become a a C 0, q q C 0 (13) The perturbed macro-frequency can be expressed as 1, where = is determined by a= and q=. The frequency shift has been well studied by the experiment and the pseudopotential well approximation theory [18, 30, 36]. For q 0.4, the relation between and q can be expressed as: q (14) Then the frequency shift is given by: q 3 0 q 0 4 C 0 (15) Equation 15 shows the frequency shift is proportional not only to the octopole component directly, but also to the square of the initial amplitude of the ions. For large values of q, the relation between and a, q is determined by Mathieu equation. The frequency shift can be expressed by: (a, q ) 0 (a, q) The first-order approximation equation becomes: d u 1 d (a q cos )u 1 (16) 1 4 (a q cos ) 3 C 0 3 cos 3 0 (17)

4 J Am Soc Mass Spectrom 010, 1, CHARACTERISTICS IN NONLINEAR ION TRAP MS 1591 where a q cos represents the oscillation frequency. It can be replaced by a shorter form of. From eq 17, we have: d u 1 d u C 3 0 cos 3 (18) The particular solution of eq 18 is u C 3 0 cos 3 (19) So the ion trajectory becomes: u u 0 u 1 C n cosn S n sinn C 0 cos 3 (0) Results and Discussion As eqs 8 to 13 show, the motional characteristics of trapped ions not only depend on Mathieu parameters a and q originally but also relate to the percentage of the octopole field, and the initial amplitude of the ions, C 0. Ions with the same m/z but different initial amplitude or velocity in the same trap would be ejected in different time, some of which are stable while others are unstable. When ejected with a mass-selective instability scan, the ions with the same m/z will not be ejected synchronously and this is an important factor influencing the mass resolution. Equation 0 reveals that because of the nonlinear field, the trajectory of ions deviates from linear and reflects a nonlinear frequency shift 3=. Equation 13 shows that as the rf field increases nonlinearly in the r- and z-direction with distance from the field center, which is true for the stability boundary q eject, the deviation from linearity is the stronger, the farther the location of the ion from the center. The ions in the center of the nonlinear ion trap will only show the characters of linear part. Once the ions are cooled by a damping gas into a small cloud around the center, they would be trapped just as they are in an ideal quadrupole trap. Figure 1a shows the shift of the stability boundary q eject with a 0, which corresponds to 1 for C 0 r 0 versus the nonlinear parameter. Figure 1b shows the shift of the stability boundary q eject with a 0 for 0.3 versus the ions initial amplitude C 0. Figure 1c) shows the shift of the stability boundary q eject with a 0 for 0.3 versus the ions initial amplitude C 0. Owing to the effect of the octopole field, macrofrequency of the ions increases with the enhancement of the positive octopole component while the frequency decreases with negative octopole component (Figure 1a). Similar results have been demonstrated by Douglas and Menon [18, 36]. It is known that the stability boundary is at q in an ideal trap (corresponding Figure 1. Plot of q value of mass-selective instability ejection in the stability boundary versus (a) positive and negative octopole superposition, (b) initial amplitudes for positive octopole superposition, and (c) initial amplitudes for negative octopole superposition. to 0inFigure 1a). Considering that the q eject, which deviates from the original q due to the octopole component, the original stability boundary q can be calculated by substituting q into eq 13. So a clear relation between q and, C 0 can be obtained by

5 159 ZHOU ET AL. J Am Soc Mass Spectrom 010, 1, q eject C 0. As the stability boundary q eject is relevant with the initial amplitude, the positive octopole fields will lower the q eject as the oscillation amplitude increases. Once the ion reaches the instability boundary q eject and becomes unstable, it will take up energy and increase its amplitude, the ejection process will be accelerated and the resolution will be improved [34] (Figure 1b). The negative octopole fields will delay the ejection and lead to poor mass resolution (Figure 1c) [37, 38]. Figure depicts a plot of versus q, obtained from different coefficients of octopole showed in Figure a and initial amplitude with octopole field 0.3 and 0.3 showed in Figure b and c, respectively. As shown in Figure, the data points match closely with the ideal quadrupole ion trap values under the small octopole percentage (the curve for 0.03 in Figure a) or initial amplitude of ions (the curve for C 0.5 in Figure b, c). However, clear deviation occurs when the values of octopole percentage and initial amplitude of ions increase. Generally, as the effect of misalignment of the trap, truncation of electrodes [1] etc., the field becomes nonlinear, and the octopole superposition in quadrupole ion trap is about 0.03 [1, 39]. For the sake of small size and easy fabrication, the hyperboloid electrodes in quadrupole ion trap should be replaced by a cylindrical ring electrode and two planar end-cap electrodes. The cylindrical geometry would lead to highly octopole superposition as high as 0.3 [40 44]. To discuss the effect induced by the nonlinear field, we take the following nonlinear parameter induced by octopole component for example, C 0 r 0 and A 4 /A 0.03, 0.3, respectively. Using the method of normalization, we have r 0 1 and 0.03, 0.3, respectively. Figure 3a shows the shift of boundary in the lowest stability region (the region between 0 and 1) versus the Mathieu parameter, a and q, after normalization. Here the stability regions for practical traps have been compared with that for an ideal quadrupole trap. Figure 3b shows the shift in the lowest stability region in detail. Figure 3 reveals that the stability boundary with a 0 will move to negative direction of q and it decreases as the positive octopole field increases. An opposite occasion is shown for negative octopole field. For normal operation of the quadrupole mass filter [45, 46], one holds the U/V ratio (that is, the a/q ratio) constant and scans both U and V. A fixed a/q ratio implies a straight line on the a-q plane, and this operation line intersects the stability region at two points, only the ions lying between the two points are stable. Because of the nonlinear field, the stability region deviates from the ideal case. Using this scan mode, the two intersection points also deviate from the ideal value. Consequently, it will affect the mass resolution. Figure. Plot of versus q along the a 0 axis of the Mathieu stability region for three-dimensional quadrupole ion trap under different (a) octopole superpositions, the solid curves represent different values of octopole component: 0 (red), 0.03 (blue), 0.3 (green), and 0.3 (orange), (b) initial amplitudes of ions with positive octopole field 0.3, (c) initial amplitudes with negative octopole field 0.3, the solid curves represent different values of initial amplitude C 0 (red), 0.5 (blue), and 0.5 (green). The curve is calculated from the continued fractions of as a function of q for an ideal quadrupole ion trap [1].

6 J Am Soc Mass Spectrom 010, 1, CHARACTERISTICS IN NONLINEAR ION TRAP MS 1593 boundary of the end cap electrode at z r 0. Figure 4b shows the frequency in the ideal ion trap and the nonlinear trap with positive octopole field with a 0 and q 0.. Under these conditions, the nonlinear effect is not significant, the pseudopotential well approximation [18, 36] as well as the numerical methods [37, 38] are widely used to get the secular frequency and the oscillation trajectories of ions. Figure 4b clearly shows the trajectories in the ideal and the nonlinear quadrupole trap. The frequency shift can be reflected by Figure 3. The lowest Mathieu stability region for threedimensional quadrupole ion trap under different octopole superposition: (a) for z-motion, (b) for r -and z-motions. The solid curves represent the stability regions in quadrupole ion trap with different values of octopole component: 0 (red), 0.03 (blue), 0.3 (green), and 0.3 (purple). For positive ions, the resolution will be improved dramatically if the added octopole field is positive. On the contrary, the resolution will be reduced deeply when the added field is negative [30]. Figure 4 shows the ion trajectories in linear and nonlinear trap, which have different initial amplitudes in the quadrupole field, coupled with positive octopole component. The eqs 13, 15, and 16 reveal that the frequency of the ions depends on both the nonlinear coefficient and the initial amplitude of the ions. As both the positive octopole component and the initial amplitude increase, ions gradually move out of the stability region. Figure 4a shows that the ions are stable in the ideal quadrupole trap ( 0) with a 0, q 0.9. However, for the nonlinear ion trap ( 0.3), only the small initial amplitude ions such as C 0 r 0 10 (q q C ) remain in the stability region. For large initial amplitude, C 0 r 0, the amplitude of the ions will increase exponentially and exceed the Figure 4. Ion trajectory with different Mathieu parameters (a) q 0.9, the solid curves represent ion trajectories in an ideal ion trap (red), nonlinear ion trap with 0.3, C 0 r 0 10 (blue), and 0, C 0 r 0 (green), (b) q 0., the solid curves represent ion trajectories in an ideal ion trap (red) and nonlinear ion trap with 0.3 (blue), (c) is the Fourier transformation from (b).

7 1594 ZHOU ET AL. J Am Soc Mass Spectrom 010, 1, the trajectories. Figure 4c shows the secular frequency by taking Fourier transformation of the ions trajectory, where the x-axis unit represents the Mathieu parameter. It is obvious C 0 C C 4... and the deviation of the macro-motion frequency 1 and the micro-motion frequency 1 become clear as octopole is of superposition. Conclusion The direct solutions of nonlinear Mathieu equation is proposed by using PLK method, an analytical expression on perturbed a and q can be obtained and the Mathieu parameter could thus be calculated and used for the calibration of the stability boundary. At the appearance of octopole potential, the frequency deviates from the ideal condition and generates a nonlinear frequency shift. Consulting the method of pseudopotential well approximation, the modified = could be used to present the macro-motion frequency, and the nonlinear equation degenerates to a linear Mathieu equation. This method could be useful to solve the problems on other multipole effects, stability boundary effects and resonance ejection with octopole field etc. Acknowledgments The authors thank Joe Ding and Professor H. W. Chen for proofreading the manuscript. The authors acknowledge support for this work by the National Natural Sciences Foundation of China under grant nos and Appendix A For solving Duffing equation: d u dt 0(u u 3 ) 0 (a) The first step of PLK method is to introduce a new dimensionless variable t and eq (a) becomes: (b) 0 1 Where u 0, 0, and u, are the amplitude, secular frequency of the linear and nonlinear oscillation respectively, and is a dimensionless variable which relate linear frequency 0 and nonlinear frequency (due to the cubic term). Substituting equation (b, d, and e) into Duffing eq (a) and comparing the order of, the relation between 0 and can be obtained. ( 0 1 ) d (u 0 u 1 ) d 0 [(u 0 u 1 ) (u 0 u 1 ) 3 ] 0 In the nonlinear Mathieu equation: d u d (a q cos )(u u3 ) 0 The secular frequency of the linear oscillation 0 in eq (a) becomes time-dependent (a-qcos) in eq (g). General treatment is using a time-average method (pseudopotential well approximation) to make frequency independent on time, and the nonlinear Mathieu equation become Duffing equation. In this paper, a modified PLK method is proposed. The first step of the modified PLK method is also to define a new variable, however, as the frequency 0 does not appear in eq (g), a direct expansion as eq (e) becomes unavailable. An assumed frequencydependent variable is induced to represent the frequency characters. Adopting a similar definition As the induced variable and t are both dimensionless variables, the here does not have a physical meaning of secular frequency which is different from the in Duffing equation. The relates and 0 (initial condition) in Duffing equation, however, the initial condition is a 0 and q 0 and the relates a, q, and a 0, q 0 in Mathieu equation. Comparing the definition (h) with (b), the is a dimensionless variable which represents the ratio of the linear and nonlinear secular frequency. The proceeding expansion can be expressed as: u u 0 u 1 (e) (f) (g) (h) (i) u d d 0(u u 3 ) 0 (c) The second step is to expand u and as asymptotic series in terms of : u u 0 u 1 (d) 0 1 Substituting eqs (h), (i), and (j) into eq (g) ( 0 1 ) d (u 0 u 1 ) d (a q cos )[(u 0 u 1 ) (u 0 u 1 ) 3 ] 0 (j) (k)

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