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.