Xiong Caiqiao, Zhou Xiaoyu, Zhang Ning, Zhan Lingpeng, Chen Yongtai, Chen Suming, Nie Zongxiu
Beijing National Laboratory for Molecular Sciences, Key Laboratory of Analytical Chemistry for Living Biosystems, Institute of Chemistry, Chinese Academy of Sciences, Beijing, 100190, China.
J Am Soc Mass Spectrom. 2015 Aug;26(8):1338-48. doi: 10.1007/s13361-015-1145-7. Epub 2015 Apr 30.
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 frequency in the superimposed octopole field, ε, were obtained by solving the nonlinear Mathieu equation (NME). The obtained accuracy of the HB method was comparable with that of the 4th RK method at the Mathieu parameter, q = 0.6, and the applicable q 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 frequency shift, Δβ, and ion amplitude variation, Δ(C(2n)/C0) (n ≠ 0). New phenomena regarding Δβ were observed, although extensive studies have been performed based on the pseudo-potential well (PW) model. For instance, the |Δβ| at ε = 0.1 and ε = -0.1 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)/C0) (n ≠ 0) are studied, and the associated study has been a challenge for both theoretical and numerical methods. The nonlinear effects of Δ(C(2n)/C0) (n ≠ 0) and Δβ were found to share some similarities at q < 0.6: both of them were proportional to ε, and the square of the initial ion displacement, z(0)(2).
与数值方法相比，非线性保罗阱中离子运动的理论表征总是存在精度低和适用性差的问题。为了克服这些困难，开发了理论谐波平衡（HB）方法，并通过数值四阶龙格 - 库塔（4阶RK）方法进行了验证。使用HB方法，通过求解非线性马蒂厄方程（NME），得到了叠加八极场ε中离子的解析轨迹和离子运动频率。在马蒂厄参数q = 0.6时，HB方法获得的精度与4阶RK方法相当，并且适用的q值可以扩展到整个第一稳定区域，精度令人满意。研究了离子运动的两种非线性效应，包括离子频率偏移Δβ和离子振幅变化Δ(C(2n)/C0)（n≠0）。尽管已经基于伪势阱（PW）模型进行了广泛研究，但观察到了关于Δβ的新现象。例如，发现在ε = 0.1和ε = -0.1时|Δβ|不同，但在PW模型中它们是相同的。这是首次研究关于Δ(C(2n)/C0)（n≠0）的非线性效应，并且相关研究对理论方法和数值方法来说都是一个挑战。发现在q < 0.6时，Δ(C(2n)/C0)（n≠0）和Δβ的非线性效应有一些相似之处：它们都与ε以及初始离子位移z(0)(2)的平方成正比。
