S. Yan1,2, M. Rosenbusch2*and Z. Huang1**
1Institute of Mass Spectrometry and Atmospheric Environment, Jinan University, Guangzhou 510632, China
2Wako Nuclear Science Center (WNSC), Institute of Particle and Nuclear Studies (IPNS), High Energy Accelerator Research Organization (KEK), Wako, Saitama 351-0198, Japan
E-mail addresses:rosmar@post.kek.jp(M. Rosenbusch *), hzx126@126.com(Z.Huang**)
Abstract :
RATIONALE : The multi-reflection time-of-flight mass spectrograph (MRTOF-MS) is a complex nonlinear system with dozens of variables that are impossible to determine in theory. Numerical analysis is the only method to determine a solution. Therefore, a numerical simulation is applied with a modified Nelder–Mead simplex (MNMS) algorithm for optimizing voltage configurations.
METHODS : Ion trajectories for injection and confinement are simulated using the software SIMION 8.1. The goal of optimization is to find a more suitable configuration for the electric field. This task becomes more challenging as the number of variables, the complexity of the objective function, and the accuracy of the variable intervals increase. A simplex search algorithm was used to perform the optimization process. We modified the searching algorithm by incorporating a variable transformation to ensure that the variables have smooth boundaries. Additionally, we introduced a dedicated benchmark to facilitate global searches.
RESULTS : By iteratively using the MNMS algorithm, a total of eight electrodes have been optimized, resulting in a smaller beam size and more efficient ion transport.
CONCLUSIONS : The MNMS algorithm is effectively for optimizing nonlinear MRTOF-MS system. It improves the adaptability and globality of the original algorithm, making it applicable for the numerical analysis of complex mass spectrometry systems and problems in engineering.
Key words: Multi-reflection time-of-flight mass spectrograph, numerical analysis, ion transport optimization, modified Nelder–Mead simplex algorithm, constrained nonlinear problem, minimizer solver.
Introduction
The multi-reflection time-of-flight mass spectrograph (MRTOF-MS), first proposed 30 years ago [1], is a fast and precise technique to measure the masses of ions. It has rapidly gained favor at radioactive ion beam (RIB) facilities for high-precision mass measurements of radioactive nuclides, such as CERN-ISOLDE (Switzerland) [2], RIKEN-RIBF(Japan) [3, 4], GSI (Germany) [5], and others (see references in [9]). A new MRTOF-MS for nuclear mass measurements has been constructed at the SLOWRI facility at RIKEN-RIBF [9]. This new structure is described with technical details and features in [6], wherein the design is similar to the previous apparatus reported in Ref[4]. For the initial operation, the voltage configuration previously used for the electrostatic mirrors in the older apparatus has been applied to the new set-up. The potentials described in Ref[4] have been determined through a differential algebra simulation and were optimized with the assumption that the ions start in the center of the MRTOF device.
MRTOF-MS is capable of achieving mass resolving powers exceeding 105 and measurement durations on the order of milliseconds [7], enabling a folded ion trajectory for a flight distance typically in the range of a few hundred meters. To achieve optimal performance, it is crucial to properly adjust the distribution of the electric field and inject ions as a focused ion pulse. This means minimizing the uncorrelated (thermal) energy spread, reducing the radial spatial distribution (in the case of concentric systems), and aligning the ion beam with the optical axis of the MRTOF-MS. So, our focus was on optimizing ion transport, which had previously only been achieved through experimentation with scanning voltage settings. But it is a time-consuming process. Therefore, numerical simulations are used, and an algorithm is applied to automatically search for optimized lens voltages that minimize the beam spot size at multiple positions.
The Nelder–Mead simplex (NMS) algorithm, originally published in 1965 by Nelder and Mead [8], is a well-known direct search algorithm for finding local minima. It does not require any information about the derivatives of the function, making it suitable for most common problems in science and engineering. On the other hand, the original Nelder–Mead algorithm is designed to solve unconstrained problems, which means that there are no limitations on the input variables. This can pose a challenge in engineering, as there are always variable constraints to consider, such as upper voltage limits in an ion-optical apparatus. Bound constraints are used to limit the size of each variable, thus excluding solutions that have no physical meaning.
In this study, we modified the NMS algorithm and utilized a variable transformation suggested by Nelder and Mead to automatically enforce constraints within a specific range. Meanwhile, the original local minimum was improved to become a global minimum by the re-check function. We provide a brief description of our apparatus in Section 2 and a numerical simulation setting in Section 3. The searching algorithm, modified with constraints, is introduced in Section 4. The results and discussion of the simulation are presented in Section 5, followed by a summary and an outlook for this auto-search algorithm.
Apparatus
Our implementation of the new MRTOF-MS at the SLOWRI facility uses a suite of radio-frequency (RF) ion traps and three lenses with two pairs of steerers, a pair of electrostatic ion mirrors with a single refocusing lens (two lenses available), and a long field-free drift region between the mirrors, as shown in Fig. 1. In this work, we primarily focus on the electrodes of the magnified area in Fig. 1, which exert a significant influence on the ion beam during its transfer to the MRTOF-MS, i.e. A1 to A3, and DT1 to DT3. Furthermore, we will focus on the trajectories inside the MRTOF device. Geometric details, functionality, and the timing structure can be reviewed in Refs [4, 6, 9]. The mirrors confine the ions so that they are reflected back and forth and separate by mass with their increasing flight path increasing with time. Electrostatic lenses are used for fine tuning of the transportation and confinement of the ions, more specifically, tuning of the ion pulse profile.