Analytical Approach for Description of Ion Motion in Quadrupole Mass Spectrometer
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Analytical Approach for Description of Ion Motion in Quadrupole Mass Spectrometer Vladimir I. Baranov MDS SCIEX, Concord, Ontario, Canada Implementation of the analytical method of the solution of the Mathieu equation in conjunc- tion with the algebraic presentation of Mathieu functions is discussed in this work. This approach is used for the analytical expression of fundamental properties of the quadrupole field such as ion trajectory stability and transmission. Extensive comparison with the matrix method is presented with demonstration of the fundamental advantages of the analytical method. However, contrary to the matrix method, the analytical method is limited to the cos trapping waveforms. (J Am Soc Mass Spectrom 2003, 14, 818–824) © 2003 American Society for Mass Spectrometry he recent development of algebraic methods to trajectory over only one cycle. The major limitation of compute Mathieu functions [1–3] has a potential this method is associated with the absence of an ana- Tto simplify significantly the theoretical descrip- lytical expression for the ion trajectory and, as a result, tion of the motion of ions in a quadrupole trap. As a the necessity to involve the numerical method for result of this progress the algebraic aspects of Mathieu averaging of the ion beam properties in the presence of functions were implemented in computer algebra sys- the trapping potentials and collisions. The matrix tems such as Mathematica [4] covering a broad range of method is used in the present work for direct compar- a and q parameters. In addition to the obvious conve- ison. The analytical method allows the analytical for- nience of the analytical method of solution of the mulation of an ion trajectory over an unlimited number Mathieu equation in conjunction with the algebraic of cycles and does not need the matrix method. How- presentation of Mathieu functions, this approach allows ever, many achievements of the matrix method could be a single solution to be used on a complete ion trajectory. incorporated. Here, simple expressions for the stability The closed formulae obtained provide an alternative conditions, acceptances, and resonances are presented. method to the numerical solution of the Mathieu equa- As a weakness of the analytical method of solution of tion and to the matrix method. This approach also offers the Mathieu equation in conjunction with the algebraic some possibilities for analytical expression of funda- presentation of Mathieu functions it should be noted mental properties of the quadrupole field such as ion that the method is still limited to the cos trapping trajectory stability, transmission, and momentum/en- waveforms. ergy characteristics of ion motion. Despite its universal power, the numerical solution of the Mathieu equation has its inherited limitations: It Equations of Ion Motion is difficult to distinguish between instability of a par- Solutions to the Mathieu differential equation ticular ion trajectory and mismatch between the ion source emittance and the quadrupole acceptance; accu- d2u mulation of the computational error limits the ion ϩ ͑␣ Ϫ 2qcos͑2͑ ϩ ͒͒͒u ϭ 0 (1) d2 0 residence time; the numerical solution offers limited insight even in the most powerful Monte-Carlo form. can be presented in terms of the Mathieu functions as The stability of the ions’ motion and their acceptance in follows: an ideal quadrupole field with periodic time varying potentials was intensively investigated by the matrix ͑͒ ϭ ͑ ϩ ͒ ϩ ͑ ϩ ͒ u K1C a, q, 0 K2S a, q, 0 method (for example [5, 6]). It is not limited to cos , (2) ͑͒ ϭ ˙ ͑ ϩ ͒ ϩ ˙ ͑ ϩ ͒ trapping waveforms and can be used directly for solu- u˙ K1C a, q, 0 K2S a, q, 0 tion of the more general Hill equation. The matrix method requires a numerical method to calculate an ion where K1,2 are the integration constants. The even Mathieu function with real characteristic value a and ϩ Published online June 25, 2003 parameter q is denoted C(a, q, 0), the odd solution Address reprint requests to Dr. V. I. Baranov, MDS SCIEX, 71 Four Valley ϩ Drive, Concord, Ontario L4K 4V8, Canada. E-mail: vladimir.baranov@ by S(a, q, 0) and their derivatives with respect to the ϩ ϩ sciex.com 0 variable (can be complex) are C˙ (a, q, 0) and © 2003 American Society for Mass Spectrometry. Published by Elsevier Inc. Received January 7, 2003 1044-0305/03/$30.00 Revised March 18, 2003 doi:10.1016/S1044-0305(03)00325-8 Accepted April 11, 2003 J Am Soc Mass Spectrom 2003, 14, 818–824 ANALYTICAL DESCRIPTION OF ION MOTION 819 ϩ S˙(a, q, 0), respectively. The solution of the Mathieu equation for given a, q, , 0, and initial conditions (ui, u˙ i) can be presented as follows: 1 u͑͒ ϭ ͓u˙ Q͑a, q, , ͒ ϩ u L͑a, q, , ͔͒ P͑a, q͒ i 0 i 0 , 1 u˙ ͑͒ ϭ ͓u˙ Q˙ ͑a, q, , ͒ ϩ u L˙ ͑a, q, , ͔͒ P͑a, q͒ i 0 i 0 (3) where ͑ ͒ ϭ ˙ ͑ ͒ ͑ ͒ Ϫ ͑ ͒ ˙ ͑ ͒ P a, q C a, q, 0 S a, q, 0 C a, q, 0 S a, q, 0 ͑ ͒ ϭ ͑ ϩ ͒ ͑ ͒ Q a, q, , 0 C a, q, 0 S a, q, 0 Ϫ ͑ ͒ ͑ ϩ ͒ C a, q, 0 S a, q, 0 Figure 1. Ion trajectory calculated according to eqs 3, 5 is ϭ ϭ ϭ presented for (ui, u˙ i) (0.15, 0.1) and a 0, q 0.4 together with the ellipse ϭ n for ϭ 1/5. ͑ ͒ ϭ ˙ ͑ ͒ ͑ ϩ ͒ 0 L a, q, , 0 C a, q, 0 S a, q, 0 Ϫ ͑ ϩ ͒ ˙ ͑ ͒ C a, q, 0 S a, q, 0 For any vector of initial conditions (ui, u˙ i) we have: and ͑ ͒ ϭ ⅐ ͑ ͒ ϭ ͩ m11 m12 ͪ ⅐ ͑ ͒ ϭ ui, u˙ i M ui, u˙ i ui, u˙ i 1 m21 m22 K ϭ ͑Ϫu˙ C͑a, q, ͒ ϩ u C˙ ͑a, q, ͒͒ 1 P͑a, q͒ i 0 i 0 Ά 1 1 ͓ ͑ ͒ ϩ ͑ ͔͒ K ϭ ͑u˙ S͑a, q, ͒ Ϫ u S˙ ͑a, q, ͒͒ ͑ ͒ u˙ iQ a, q, , 0 uiL a, q, , 0 , 2 P͑a, q͒ i 0 i 0 P a, q (5) 1 u˙ i ͓u˙ Q˙ ͑a, q, , ͒ ϩ u L˙ ͑a, q, , ͔͒ K ϭ ͑ ͒ i 0 i 0 1 ˙ ͑ ͒ P a, q S a, q,0 ϭ or for 0 0. Ά ui K ϭ If one limits the time variable to n values (n complete 2 C͑a, q,0͒ RF periods) and assumes that 0 is zero, then matrix M is equal to the transfer matrix. The matrix method Eq 2 is essentially the same as eq 3 and both are equally requires a numerical method to calculate an ion trajec- useful. However, eq 3 has some advantages in associa- tory over one cycle if 0 0. These limitations are not tion with the matrix method. required in the analytical approach and eqs 4 and 5 are It should be recognized that the analytical method of sufficient for any and 0. In Figure 1 the solution (see ϭ ϭ ϭ solution of eq 1 does not require anything else to be eq 3) is presented for (ui, u˙ i) (0.15, 0.1) and a 0, q ϭ complete. However, for evaluation of the analytical 0.4 together with the ellipse n for 0 1/5 . Intersec- method it is beneficial (but not necessary) to follow the tions between the ellipse and trajectory happen every matrix method formalism (see, for example [5, 6]), ϭ n, as should be. which is based on a pair of independent solutions for ϭ initial conditions (ui, u˙ i) (1,0): Stability Parameterization According to the Liouville theorem: ͑ ͒ ˙͑ ͒ ͑͒ ϭ L a, q, , 0 ϭ ͑͒ ϭ L a, q, , 0 ϭ u m11; u˙ m21, P͑a, q͒ P͑a, q͒ C˙ ͑a, q, ϩ ͒S͑a, q, ϩ ͒ Ϫ C͑a, q, ϩ ͒S˙͑a, q, ϩ ͒ 0 0 0 0 ϭ 1 ˙ ͑ ͒ ͑ ͒ Ϫ ͑ ͒˙͑ ͒ C a, q, 0 S a, q, 0 C a, q, 0 S a, q, 0 and for ͑u , u˙ ͒ ϭ ͑0,1͒: (4) i i (6) ͑ ͒ ˙ ͑ ͒ Q a, q, , 0 Q a, q, , 0 This can be true only if the nominator and denominator u͑͒ ϭ ϭ m ; u˙͑͒ ϭ ϭ m . P͑a, q͒ 12 P͑a, q͒ 22 are independent of the time variable and initial phase 820 BARANOV J Am Soc Mass Spectrom 2003, 14, 818–824 for n ϭ 1: 1 m ϩ m ϭ 2cos͑͒ ϭ 11 22 P͑a, q͒ ͑ ͒ ˙ ͑ ͒ Ϫ ˙ ͑a, q,0͒S͑a, q, ͒ ϫ ͩ C a, q, S a, q,0 C ͪ, ϩ C͑a, q,0͒S˙ ͑a, q, ͒ Ϫ C˙ ͑a, q, ͒S͑a, q,0͒ 1 1  ϭϮ arccosͩ ͑m ϩ m ͒ͪ 2 11 22 1 C͑a, q, ͒ ϭ arccosͩ ͪ C͑a, q,0͒ 1 S˙ ͑a, q, ͒ ϭ arccosͩ ͪ (8) S˙ ͑a, q,0͒ Although all functions in M formally depend on the initial phase 0 and the time variable , the matrix ϭ ϭ method defines 0 0 and . The stability number  ϭ ϭ is defined for 0 0 and and is complex for unstable trajectories.