RESEARCH ARTICLE. A Theoretical Method for Characterizing Nonlinear Effects in Paul Traps with Added Octopole Field

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1 B American Society for Mass Spectrometry, 215 J. Am. Soc. Mass Spectrom. (215) 26:1338Y1348 DOI: 1.17/s RESEARCH ARTICLE A Theoretical Method for Characterizing Nonlinear Effects in Paul Traps with Added Octopole Field Caiiao iong, 1 iaoyu Zhou, 1 Ning Zhang, 1 Lingpeng Zhan, 1 Yongtai Chen, 2 Suming Chen, 1 Zongxiu Nie 1,3 1 Beijing National Laboratory for Molecular Sciences, Key Laboratory of Analytical Chemistry for Living Biosystems, Institute of Chemistry, Chinese Academy of Sciences, Beijing, 119, China 2 School of Information Engineering, Wuhan University of Technology, Wuhan, 437, China 3 Beijing Center for Mass Spectrometry, Beijing, 119, China Abstract. In comparison with numerical methods, theoretical characterizations of ion motion in the nonlinear Paul traps always suffer from low accuracy and little applicability. To overcome the difficulties, the theoretical harmonic balance (HB) method was developed, and was validated by the numerical fourth-order Runge-Kutta (4th RK) method. Using the HB method, analytical ion trajectory and ion motion freuency in the superimposed octopole field, ε, were obtained by solving the nonlinear Mathieu euation (NME). The obtained accuracy of the HB method was comparable with that of the 4th RK method at the Mathieu parameter, =.6, and the applicable values could be extended to the entire first stability region with satisfactory accuracy. Two sorts of nonlinear effects of ion motion were studied, including ion freuency shift, Δβ, and ion amplitude variation, Δ(C 2n /C )(n ). New phenomena regarding Δβ were observed, although extensive studies have been performed based on the pseudo-potential well (PW) model. For instance, the Δβ atε=.1 and ε=.1 were found to be different, but they were the same in the PW model. This is the first time the nonlinear effects regarding Δ(C 2n /C )(n ) are studied, and the associated study has been a challenge for both theoretical and numerical methods. The nonlinear effects of Δ(C 2n /C )(n ) and Δβ were found to share some similarities at <.6: both of them were proportional to ε, and the suare of the initial ion displacement, z() 2. Key words: Paul trap, Nonlinear, Harmonic balance, Runge-Kutta, Higher-freuency harmonics Received: 11 January 215/Revised: 18 March 215/Accepted: 19 March 215/Published Online: 3 April 215 Introduction Three-dimensional (3D) uadrupole ion (Paul) trap, developed by Paul and Steinwedel in 1953 [1], opens a new era of ion trap mass spectrometry (MS) [2 5]. Subseuently, a series of Paul trap s geometrical variants, including 3D cylindrical [6], 2D rectilinear [7], 2D linear [8, 9], 2D toroidal [1], Electronic supplementary material The online version of this article (doi:1.17/s ) contains supplementary material, which is available to authorized users. Correspondence to: Zongxiu Nie; znie@iccas.ac.cn and 2D halo [11], etc., were developed, which make the family of the Paul traps even more powerful. In contrast to sectors [12, 13] and time of flights [14, 15], the Paul traps have a uniue feature: the sample ions are confined in a fixed space during the mass analysis. This intrinsic advantage can be readily utilized to perform tandem MS (MS/MS) experiments, which is highly valued in chemical analysis for structure determination [16, 17]. Currently, the Paul traps have been widely used in laboratory-scale instruments, such as LCQ [18], LTQ (Thermo Scientific, San Jose, CA, USA) [19], and QTRAP (AB Sciex, Concord, Canada) [9], and portable instruments, such as Mini 12 (Purdue University, West Lafayette, IN, USA) [2], ChemSense 6 (Griffin Inc., West Lafayette, IN, USA) [21], and Tridion-9 (Torion Technologies Inc., American Fork, UT,

2 C. iong et al.: Nonlinear Effects in Paul Trap 1339 USA) [22]. The Paul trap instruments play important roles in a variety of applications, such as food safety [23], explosive detection [24], environmental monitoring [25], drug analysis [26], and space exploration [27]. In a practical Paul trap, its internal electric fields are comprised of a dominant linear uadrupole component and other nonlinear higher-order multipole ones, such as hexapole and octopole [2]. The nonlinear fields are important for the performance of the trap, in terms of mass resolution, sensitivity, and mass accuracy. At first the nonlinear fields are undesirable but are inevitably introduced because of the assembly misalignment [28], fabrication precision [29], and truncation of electrodes [3], etc. Moreover, the nonlinear fields are regarded as the main factor in the degradation of the ion trap performance. For instance, the resultant nonlinear effects, such as ion delay ejection [31, 32], nonlinear resonance [33, 34], and freuency shift [35, 36], are considered to be harmful to the mass resolution, sensitivity, and mass accuracy, respectively. With the in-depth study, the nonlinear fields have been found to be beneficial and have the potential to break the performance barrier and develop new operation modes of the ion trap. Such examples can be readily found in, e.g., Finnigan and Bruker-Franzen Paul traps [37], cylindrical and rectilinear Paul traps [6, 7], and axial ejection in QTRAP [9]. Other than the ion trap, the study of nonlinear field is also applicable to ion funnel [38], ion guide [39], ion manipulator [4], ion collision cell [41], mass filter [42], Orbitrap [43], and Penningtrap[44]. Ion motion in the uadrupole field with superimposed nonlinear field is described by the nonlinear Mathieu euation (NME) [45], which can be solved by using both theoretical methods and numerical methods. In comparison with the numerical methods, the theoretical methods can provide better understanding of the associated nonlinear effects of ion motion. However, the current theoretical methods still have limitations, including relatively low accuracy and little applicable range. For example, regarding ion trajectory calculations, the pseudo-potential well (PW) method is typically valid for Mathieu parameters, <.4 [46]. Even so, the calculated ion trajectory still has a low accuracy. These limitations will hinder the applications on the design of ion trap instruments. For instance, it is common knowledge that at high values, e.g., >.6, the high-freuency harmonics of ion motion would have a significantly large impact on the ion trajectory and the trap performance. However, because of the restrictions of the available methods, the associated nonlinear study has never been performed. The limitations of the theoretical methods can be largely attributed to the approximation treatments, which have to be employed to make the NME analytically solvable. Generally, two sorts of approximation treatments are most often employed, including the PW method [46] and the perturbation methods. The latter include a variety of methods, such as Poincare-Lighthill-Kuo (PLK) [45], harmonic balance (HB) [47 49], Lindstedt- Poincare [5 54], asymptotic [55], homotopy [56, 57], and multiple time scales [58, 59]. In most cases, using the perturbation method alone is still difficult to solve the NME; therefore, the perturbation method and the PW method are always used together [48 59]. However, the employed PW method neglects all the higher-freuency harmonics of ion motion, which gives rise to low accuracy and little applicability. To improve the theoretical methods, it is necessary to improve or reduce the employed approximation treatments. For instance, the PLK perturbation method without using the PW model has been utilized to solve the NME [34, 39, 45, 6]. This method can extend the applicable range of Mathieu parameters to the second stability region, e.g., at a=2.7 and =3.85, with a typical ion motion freuency error of less than 15% [6]. In the PLK method, all the higherfreuency harmonics of ion motion are still considered integral, as described by the two effective Mathieu parameters, a eff and eff [45]. In other words, this is an improved approximation treatment in comparison with the PW method. For further improvement of the theoretical methods, a more accurate description of ion motion is reuired. In this article, the HB method was developed for calculating ion trajectory and ion motion freuency in the uadrupole field with a weak superimposed octopole field, ε. In the HB method, the formal solution of the ideal Mathieu euation was employed as a trial solution, in which the freuencies, 2n+β, and amplitudes, C 2n,of ion motion, including the secular harmonic (n=) and the higher-freuency harmonics (n ), were fully and accurately characterized. Theory The NME Paul trap consists of a hyperbolic ring electrode and two hyperbolic endcap electrodes. To operate the trap, an electric voltage, Φ =U Vcos(Ωt), is applied to the ring electrode with the endcap electrodes electrically grounded, where U is the

3 134 C. iong et al.: Nonlinear Effects in Paul Trap voltage of direct-current (DC), V is the voltage of radiofreuency (rf) with angular freuency Ω, and t is time. Such a configuration can generate an electric potential, which is essential for the ion trapping and mass analysis. The electric potential, Φ, can be represented by the addition of multipole components, which yields: Herein, the a and are dimensionless Mathieu parameters, is dimensionless time, and ε is dimensionless nonlinear field parameter. These dimensionless parameters are defined as: a z ¼ 2a r ¼ 8A 2eU ð5þ mr 2 Ω2 r 2 2z 2 3r Φ ¼ Φ A þ A 2 þ 4 24r 2 z 2 þ 8z 4 A 4 þ 2r 2 8r 4 ð1þ z ¼ 2 r ¼ 4A 2eV mr 2 Ω2 ð6þ where r and z represent the radial and axial coordinates, respectively; r is the radius of the ring electrode; A, A 2 and A 4 represent the dimensionless amplitudes of the monopole, uadrupole, and octopole components, respectively. The vanished hexapole component (i.e., A 3 =), indicates the trap is symmetric [3]. The r-z coupling term, 24r 2 z 2, gives rise to the coupling effect of ion motions in the r- and z-directions and the possible ion losses at the nonlinear resonance conditions [37, 61, 62]. To study the nonlinear coupling effect, the ion motion euations, also known as the NMEs, in the two directions should be solved together, which makes the complex NMEs even more difficult to be solved. For simplicity, only the ion motion along the r- or z-axis (i.e., z= or r=), respectively, was studied, where the coupling term in Euation 1 was eual to zero. Ion motion in the electric field is governed by the Newton s second law, which yields: m d2 u ¼ e Φ ð r; z Þ dt2 u u ¼ r; or z ð2þ where e is electron charge, m is ion mass. Using Euation 2, the euation of ion motion in the electric field as described by Euation 1 yields: d 2 z d 2 þ ½ a z 2 z cosð2þ z þ 2 r 2 εz 3 ¼ ð3þ ¼ Ωt 2 ε ¼ A 4 A 2 ð7þ ð8þ Euations 3 and 4 are the NMEs in the z- andr-directions, respectively. In this article, only Euation 3 was used for the illustration of the HB method, considering the ion ejection during the mass analysis was operated in the z- direction. The Euation 4 can be solved by using exactly the same method. For the case, ε=, Euation 3 becomes the ideal Mathieu euation [63], which has an exact solution of zðþ ¼ κ C 2n cosð2n þ βþ þ κ C 2n sinð2n þ βþ ð9þ where κ and κ are arbitrary constants, depending on the initial condition of the ion. C 2n and β are recursively defined constant coefficients and can be exactly solved from a and [63]; n is integer. For an initial ion velocity of zero, the κ in Euation 9 is eual to zero and the Euation 9 becomes: zðþ ¼ κ C 2n cosð2n þ βþ ð1þ d 2 r d 2 þ ½ a r 2 r cosð2þ r þ 3 εr 3 ¼ 2r 2 ð4þ Discussions about the non-zero initial ion velocity can be found in Section 1 in the Supporting Information.

4 C. iong et al.: Nonlinear Effects in Paul Trap 1341 The HB Method The HB method has already been used to solve the NME [47 49]. However, in those works, the ion secular harmonic (i.e., C term in Euation 1) is considered to have the main contribution to the ion amplitude; therefore, the higher-freuency harmonics (i.e., C 2n (n ) terms in Euation 1) are partially [47] or completely neglected [48, 49]. In practice, the C 2n (n ) terms play an increasingly important role in the ion amplitude as the value increases. For example, at =.8, there is C -2 /C =.48 in a pure uadrupole field (Table S1 in the Supporting Information). The result indicates the C 2n (n ) terms in Euation 1 should be included in the HB method. Other than the secular harmonic and higher-freuency harmonics, the ion motion in the superimposed octopole field should also include the nonlinear-freuency harmonics, e.g., with freuency, 3β [34, 45]. The amplitudes of these ion nonlinear-freuency harmonics have been found to be proportional to the octopole field strength, ε. For a given trap geometry, the ε is a small constant, which indicates that the nonlinear-freuency harmonics of ion motion can be neglected even in the high values. For these reasons, in the HB method the solution of the ideal Mathieu euation, Euation 1, isemployedas the trial solution. The trial solution is subseuently substituted into the NME, Euation 3, from which the coefficients, including the freuencies, 2n+β, and the amplitudes, C 2n, of ion motion can be determined. The procedures for deriving the β and C 2n in the NME and the ideal Mathieu euation are the same. First, substituting Euation 1 into the left-hand side terms of Euation 3, and neglecting the nonlinear-freuency terms, e.g., 3β, ityields: d 2 z d 2 ¼ κ z 3 ¼ κ 3 ¼ κ 3 4 i; j;k¼ ð2n þ βþ 2 C 2n cosð2n þ βþ ð11þ C 2i C 2 j C 2k cosð2i þ βþcosð2j þ βþcosð2k þ βþ iþ j k¼n i jþk¼n iþ jþk¼n C 2i C 2 j C 2k! cosð2n þ βþ ð12þ 2cosð2Þz ¼ κ C 2n 2 þ C 2nþ2!cosð 2n þ βþ 2cosð2Þz 3 3 ¼ κ 4 ¼ κ C 2n 2 cosð2n þ βþ ð13þ iþ j k¼n 1 i jþk¼n 1 iþ jþk¼n 1 C 2i C 2 j C 2k! cosð2n þ βþ ð14þ where i, j, andk are all integers from - to. iþ j k¼n fði; j; kþ means the summation of the functions, f(i, j, k) with all the i, j, and k satisfying i+j- k=n. Then, substituting Euations into Euation 3 yields κ 8 h i ð2n þ βþ 2 a C 2n þ C 2n 2 ε 2r 2 aκ 2 ><! ε 2r 2 κ 2 >: iþ jþk¼n iþ j k¼n 1 iþ j k¼n i jþk¼n i jþk¼n 1 iþ jþk¼n 1 9 >=! cos½ð2n þ βþ ¼ ð15þ C 2i C 2 j C 2k >; Euation 15 can only be satisfied such that 8 h i ð2n þ βþ 2 a C 2n þ C 2n 2 ε 2r 2 aκ 2 ><! ε 2r 2 κ 2 >: iþ jþk¼n iþ j k¼n 1 iþ j k¼n i jþk¼n i jþk¼n 1 iþ jþk¼n 1 9 >=! ¼ for < n < þ ð16þ C 2i C 2 j C 2k >;

5 1342 C. iong et al.: Nonlinear Effects in Paul Trap Using Euation 16,theβ and C 2n reuired by the NME s solution can be obtained, and the NME has been analytically solved. In Euation 16, it can be found that because of the superimposed octopole field (ε ), the values of β and C 2n in the superimposed octopole field would be different from those in ideal uadrupole field (ε=), which give rise to the ion freuency shift effect, Δβ,and the ion amplitude variation effect, Δ(C 2n /C )(n ), respectively. In this article, the Euation 16 was solved by using the Gauss-Seidel relaxation method [64], with a truncation to C ±8 (Table S2 in the Supporting Information). The calculated results of β and C 2n in the different superimposed octopole fields can be found in Table S3 in the Supporting Information. Results and Discussion Design of ion trap instruments can benefit from the calculation of ion trajectories, from which the mass range, resolution, sensitivity, and accuracy can be readily obtained. For calculating ion trajectories, numerical methods are generally recognized to be highly accurate. Hence, for verification of the analytical HB method, numerical results calculated by using the fourth-order Runge-Kutta (4th RK) are also shown for comparison [64]. Figure 1a shows the ion trajectories at a= and =.6 under different superimposed octopole fields: ε= (black), ε=.1 (red), and ε=.1 (blue). The 4th RK and HB results are represented by curves and dots, respectively. The ion trajectories were calculated with an initial displacement, z()=.1 r, and velocity, dz()/d=. This initial condition was used throughout the article unless otherwise specified. In Figure 1a, it can be found that the analytical HB results are in good agreement with the numerical 4th RK results. The feature can be more clearly observed in Figure 1b and c. Figure 1b is the enlarged plot of Figure 1a in the range of 62<<72. Figure 1c shows the specific differences of the.1 =.6.1 Exact, HB, HB, (c) =.6 4 th RK, 5 (d) th RK, 4 th RK, 4 th RK, = th RK.3 z err / r Figure 1. Verification of the HB method by using the 4th RK method with respect to the calculated ion trajectory. Ion trajectories at a z =, z =.6 under different octopole fields: ε= (black), ε=.1 (red), and ε=.1 (blue). The calculated results using the 4th RK method and the HB method are represented by curves and dots, respectively. The HB result at ε= (black dots) istheexact solution. The calculation time is =1 (or t=31.83 μs attherf freuency of 1 MHz). Enlarged plot of in the range of 62<<72 (or μs<t<22.91 μs). (c) The ion trajectory differences, z err, calculated by the 4th RK and HB methods in under different octopole fields: ε= (black curve), ε=.1 (red curve), and ε=.1 (blue curve). (d) Ion trajectories at a z =, z =.83, and ε=.1 using the 4th RK method (black curve) andthehbmethod(red curve). The initial displacement and velocity of the ion are set to.1 r and, respectively. This initial condition is used throughout this article, unless otherwise specified

6 C. iong et al.: Nonlinear Effects in Paul Trap 1343 two methods (i.e., z err =z RK () z HB ()), at ε= (black curve), ε=.1 (red curve), and ε=.1 (blue curve), where z RK and z HB are the displacements of ion trajectory at time,, calculated by using the 4th RK and HB methods in Figure 1a, respectively. At ε=, the ion trajectory calculated by using the HB method is also the exact solution. The little difference, z err, at ε= (black curve in Figure 1c) indicates that the 4th RK method is indeed an accurate method and can act as an Baccuracy standard,^ when the exact solutions of the NMEs are not available at ε. Hence, the same little differences, z err,atε=.1 (red curve) and ε=.1 (blue curve) in Figure 1c indicate that the HB can also accurately calculate the ion trajectories in the superimposed octopole field. In addition, using the 4th RK result as a Bstandard,^ the HB and PLK methods were compared (Figure S2 in the Supporting Information). It can be observed that both the HB and PLK methods can achieve agoodaccuracyat=.6, but the HB method is more accurate. The HB method also has a broad applicable range of for calculating ion trajectory. Figure 1d shows the ion trajectories at a=, =.83, and ε=.1 using the 4th RK method (black curve) and the HB method (red curve). At ε=.1, the employed =.83 is in the vicinity of the stability boundary, eject =.843. The shift of eject from.98 at ε= to the.843 at ε=.1 is attributable to the ion freuency shift effect [65]. Figure 1d indicates that the HB method can be applied in the entire first stability region with satisfactory accuracy. Figure 2a shows the power spectra at a= and =.6, with different superimposed octopole fields: ε= (black curve) and ε=.1 (red curve). Using fast Fourier transforms (FFTs), the power spectra were obtained from the corresponding 4th RK ion trajectories in Figure 1a with a longer calculation time, =5, (inset of Figure 2a). Among the ion motion freuency series, the ion secular freuency, β, is the fundamental one, from which all the other freuencies of ion higher-freuency harmonics can be readily obtained by using a simple euation, 2n+β. As a result, the ion secular freuency calculated by using the 4th RK method, β RK,at ε= (black curve) and ε=.1 (red curve) in Figure 2a are enlarged in Figure 2b and c, respectively.infigure2b, it can be found that at =.6 and ε=, the β RK =.4619 (black curve) matches with its exact value, β exact =.4622 (green line). The result suggests that the 4th RK method is accurate for calculating ion motion freuency and can, thus, be used to verify the HB method at ε. In Figure 2c, at=.6 and ε=.1, the calculated β HB =.4666 (green line) by using the HB method also matches β RK =.4664 (black curve). The result suggests that the HB method is at least as accurate as the 4th RK method for calculating ion motion freuency. AsshowninFigures1 and 2, the nonlinear effects of ion motion are generally too weak to be observed without an efficient tool. For example, at =.6, the magnitude of the ion amplitude (or maximum ion displacement) variation, Δz max, from ε= to ε=.1 is ca. 1 r, which is ca. 5% of the z max at ε= (Figure 1). At the same =.6, the ion secular freuency shift, Δβ, from ε= to ε=.1 is ca. 44, which is ca. 1% of the β at ε= (Figure 2). The results indicate that the good accuracy of the HB method is necessary for characterizing these nonlinear effects. First, the HB method was used to study the ion freuency shift, Δβ. In Figure 3a, the ion secular freuencies, β, along the -axis of the first stability region were calculated under different superimposed octopole fields: ε= (black curve), ε=.1 (red curve), and ε=.1 (blue curve). The ion freuency shift effect has been extensively studied based on the PW model [34, 39, 45, 47 6]; therefore, it is not a surprise to find that compared with the ion secular freuency, β, in the uadrupole field (black curve), the freuencies, β, in the positive (red curve) and negative superimposed octopole fields (blue curve) have positive and negative freuency shifts, Δβ (inset of Figure 3a), respectively. To clearly show the effect, the Δβ as a function of at ε=.1 (red curve) and ε=.1 (blue curve) is shown in Figure S3 in the Supporting Information. In the PW model [58, 59], the relative ion secular freuency shift, Δβ/β (or Δω/ω ) yields: Δω ¼ Δβ 1 ω β r 2 εzðþ 2 ð17þ 1 FFT 1 β exact =.4622 (c) β RK = β RK =.4664 β HB =.4666 β = β β z β z β z Figure 2. Verification of the HB method by using the 4th RK method with respect to the calculated ion motion freuency. Power spectra at a z = and z =.6 using fast Fourier transforms (FFTs) at ε= (black curve)andε=.1 (red curve). The inset is the ion trajectories calculated by using the 4th RK method with =5, (or t=15.92 ms at the rf freuency of 1 MHz)., (c) The numerical ion secular freuencies, β RK,at ε= and (c) ε=.1, enlarged from. The exact β exact at ε= and the analytical β HB at ε=.1 are indicated by the green lines for comparison

7 1344 C. iong et al.: Nonlinear Effects in Paul Trap (c) 8 =.6 =.8 =.4 4 (d) =.6 = [z()/r ] 2 Figure 3. The secular freuency β z as a function of z under different octopole fields: ε= (black curve), ε=.1 (red curve), and ε=.1 (blue curve). The inset is the enlarged plot in the range of 5< z <54. The relative freuency shifts, Δβ z /β, as a function of z at ε=.1 (red curve) andε=.1 (blue curve). The inset is the enlarged plot in the range of < z <2. (c) At given initial ion amplitude, z()=.1 r, Δβ z as a function of ε at different z : z =.4 (green curve), z =.6 (red curve), and z =.8 (blue curve). (d) Atgivenε=.1, Δβ z as a function of [z()/r ] 2 at different z : z =.6 (red curve) and z =.8 (blue curve) where ω is the secular freuency of ion motion and has ω=.5βω. Some references use z max instead of z() [58, 59]; however, in the PW model, it has z max =z() if dz()/d=. Euation 17 yields that at a given initial ion amplitude, z(), the magnitudes of the relative ion secular freuency shift, Δβ/β, at positive and negative superimposed octopole fields with the same field strength, e.g., ε=±.1, are the same. With high accuracy, the HB method can find more effects, which are not able to be observed in the PW model (or Euation 17). In Figure 3b, it can be found that the Δβ/β atε=.1 (red curve) is larger than that at ε=.1 (blue curve). This phenomenon could be observed for all the values (inset of Figure 3b), especially for >.6. Euation 17 also yields two linear relationships: the Δβ is proportional to (1) ε and (2) z() 2. To find the applicable range of in Euation 17, the HB method was employed to calculate the Δβ as a function of ε (Figure 3c) andz() 2 (Figure 3d) at different values. In Figure 3c, a good linear correlation between Δβ and ε can be observed at =.4 (green curve) and.1 = = Ion trajectory Secular part Higher-freuency part.3.1 =.6 = Ion trajectory Secular part Higher-freuency part Figure 4. Ion trajectories in the ideal uadrupole field at a z =, z = and z =.6. The ion trajectory (black curve) is resolved into two parts: ion secular harmonic (red curve) and ion higher-freuency harmonics (blue curve)

8 C. iong et al.: Nonlinear Effects in Paul Trap 1345 C 2 / C = = (C 2 /C ) (C 2 /C ) C -2 / C (C -2 /C ) (C -2 /C ) 1 2 = = (c) 2 = -.1 (d) = Figure 5., The relative ion amplitudes of the higher-freuency harmonics, C 2 /C and C -2 /C, as a function of z under different octopole fields: ε= (black curve), ε=.1 (red curve), and ε=.1 (blue curve). The insets of and are the enlarged plots in the range of < z <2, correspondingly. (c), (d) The relative ion amplitude variations, (c) Δ(C 2 /C )and(d) Δ(C -2 /C ), as a function of z at ε=.1 (red curve) andε=.1 (blue curve) =.6 z() r =.8 z() r (c) = (d) = [z()/r ] [z()/r ] 2-6 Figure 6., At a given value of z()=.1 r, Δ(C 2 /C )(red curve)andδ(c- 2 /C )(blue curve) as a function of ε at z =.6 and z =.8. (c), (d)atagivenvalueofε=.1, Δ(C 2 /C )(red curve)andδ(c- 2 /C )(blue curve) as a function of [z()/r ] 2 at (c) z =.6 and (d) z =.8

9 1346 C. iong et al.: Nonlinear Effects in Paul Trap =.6 (red curve), but cannot be observed at =.8 (blue curve). In Figure 3d, at =.6 (red curve) and =.8 (blue curve), a similar phenomenon can be observed with respect to the linearity between Δβ and z() 2. The decreased linearity at =.8 in Figure 3c and d indicates that the Euation 17 derived from the PW model can only be used to describe the nonlinear ion freuency shift at <.6. Generally, the PW model is considered to be valid for <.4. The extended applicable range of by using the PW model has also been found in the study of potential well depth [66]. However, the extended applicable range of in Euation 17 does not imply the PW model can be used to calculate ion trajectory at the same range. Figure 4a and b are the ion trajectories calculated in the ideal uadrupole field at = and =.6, respectively. For comparison, the ion trajectories (black curve) are resolved into two parts: one is ion secular harmonic (red curve), the other is ion higher-freuency harmonics (blue curve). At =, the ion higher-freuency harmonics have a contribution of 9.2% to the ion amplitude (Figure 4a). The weight will increase to 26.4% at =.6 (Figure 4b). The results indicate that without considering ion higher-freuency harmonics, the PW method cannot accurately calculate an ion trajectory even at low values, e.g., <.4, or study the associated nonlinear effect (i.e., the amplitude variation of ion higher-freuency harmonics), ΔC 2n (n ). The HB method was used to study the ion amplitude variation, ΔC 2n (n ). This is the first time the nonlinear effects with respect to ΔC 2n are studied. Among the ion high-freuency harmonics, C 2n (n ), the C 2 and C -2 have the largest amplitudes (Tables S1 and S3 in the Supporting Information). As a result, the amplitude variations of the C 2 and C -2 were employed as an example. Using the HB method, the relative amplitudes of C 2 /C (Figure 5a)and C -2 /C (Figure 5b) were calculated along the -axis under different octopole fields: ε= (black curve), ε=.1 (red curve), and ε=.1 (blue curve). To clearly observe the ion amplitude variation, the Δ(C 2 /C )andδ(c -2 /C ) are shown in Figure 5c and d, respectively.infigure5c and d, it is obvious that the nonlinear behaviors of Δ(C 2 /C )andδ(c -2 /C ) are different. In Figure 5d,theΔ(C -2 /C ) will monotonically increase or decrease along the -axis in the positive (red curve) or negative (blue curve) superimposed octopole fields, respectively. The observed effect of Δ(C -2 /C )issimilartothatofδβ (FigureS3inthe Supporting information). However, in Figure 5c, a transition point of Δ(C 2 /C ) around =.71 can be observed, where Δ(C 2 /C )= at both ε=.1 (red curve) and ε=.1 (blue curve). At <.71 and >.71, the Δ(C 2 /C ) will change polarity. This is different from Δβ, for which the transition point doesn t exist. Euation 16 is not easy to solve. It is desired that there is a simple euation as Euation 17 that can model the ΔC 2n (n ). If the nonlinear behaviors of ΔC 2n (n ) and Δβ are similar at given values, the modeling euation of ΔC 2n (n ) would be: ΔðC 2n =C Þ 1 ðc 2n =C Þ r 2 εzðþ 2 ; n ð18þ where (C 2n /C ) is the C 2n /C at ε=. Euation 18 includes two linear relationships: the Δ(C 2n /C ) is proportional to (1) ε and (2) z() 2. To confirm the euation, the HB method was first employed to study the Δ(C 2 /C ) (red curve) and Δ(C- 2 /C ) (blue curve) as a function of ε at a given z()=.1 r under different values: =.6 (Figure 6a) and =.8 (Figure 6b). It can be found that at =.6, both Δ(C 2 /C )(redcurve)and Δ(C- 2 /C ) (blue curve) increase linearly with ε (Figure 6a), but the good linearity disappears at =.8 (Figure 6b). Then, at a given ε=.1, the HB method was employed to study the Δ(C 2 /C ) (red curve) and Δ(C- 2 /C ) (blue curve) as a function of z() 2 at =.6 (Figure 6c) and=.8 (Figure 6d). Similarly, a good linear correlation between Δ(C 2 /C )andz() 2 (red curve) and between Δ(C- 2 /C )and z() 2 (blue curve) can be observed at =.6 (Figure 6c), but cannot be observed at =.8 (Figure 6d). Figure 6 indicates that the Euation 18 can well describe the nonlinear effects with respect to ion amplitude variation for <.6. Moreover, Figures 3 and 6 indicate that the ion freuency shift, Δβ, and ion amplitude variation, ΔC 2n (n ), have similar effects, as described in the Euations 17 and 18, respectively. Conclusion As a summary, the analytical HB method is an efficient tool for studying nonlinear effects in the Paul trap with a superimposed octopole field, ε. The accurate ion motion freuency, β, and ion harmonic amplitudes, C 2n, can be obtained by solving Euation 16. At <.6, Euations 17 and 18, as approximation euations of Euation 16, can well describe the ion freuency shift, Δβ, and amplitude variation, ΔC 2n (n ), respectively. Euations 17 and 18 describe that both the Δβ and ΔC 2n are proportional to the nonlinear field strength, ε, and the suared initial ion amplitude, z() 2.Athigher values, some interesting nonlinear effects can be observed. At >.6, the Δβ in the positive octopole field becomes significantly greater than that in the negative superimposed octopole field with the same field strength. At =.71, there is a transition point for C 2,where the ΔC 2 = even at ε. In the future, the HB method can be improved by considering the r-z coupling effects of ion motion and the nonlinear-freuency harmonics, e.g., 3β terms. Analytical expressions of β and C 2n, deduced directly from Euation 16 instead of Euations 17 and 18, are also expected for better understandings of their nonlinear effects. 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