. A new set of coefficients cijk are required each time the ion moves into a new cell (defined by the 8 grid points immediately surrounding the ion). Standard Runge-Kutta integration routines are used to compute the trajectory.
Ion trajectory simulation tools such as MacSimion are useful tools for understanding existing ion-optical designs and for evaluating new ones. Although MacSimion operates in just two dimensions, it suffices for problems with planar or cylindrical symmetry, such as an ion source with tall narrow slits, a set of quadrupole rods, or an aperture lens system. Many real systems possess no simple symmetry or have overlapping regions with different symmetries. The interface region between an ion source and a quadrupole mass analyzer, or that between the latter and an AC-only quadrupole (as in a triple quadrupole instrument) are examples which require a full three dimensional treatment.
Small 2D calculations use arrays of perhaps 100 by 200 points to represent a single plane of a problem. This often only provides a minimal rendering of the electrode geometries but is adequate to produce semi-quantitative results. The same minimal calculation in 3D requires tens of megabytes of memory, with refine times running into many hours. High performance computers using RISC processors would appear to be attractive platforms on which to tackle such simulations. A native code version of MacSimion for the PowerMac was found to out-perform earlier CISC processors by as much as a factor of five.
MacSimion 3D is a new program we have developed, allowing a system of electrodes to be set up in 3D. The memory requirements for a simulation are (4n+2) xyz bytes, where x, y and z are the dimensions and n is one greater than the number of time-dependent waveforms in the problem. Laplace's equation is solved iteratively on the three-dimensional grid by a relaxation process (refining) until satisfactory convergence is reached. To improve execution speed the procedure initially operates on a coarse grid and progresses to maximum resolution near the end of the refining operation, which typically takes about one hour for an array containing 106 grid points. The resulting potential distributions can be exported into, and examined using the 3D visualization package, Dicer.
During ion trajectory calculations we compute a 64 term interpolating polynomial for the potential which has the form . A new set of coefficients cijk are required each time the ion moves into a new cell (defined by the 8 grid points immediately surrounding the ion). Standard Runge-Kutta integration routines are used to compute the trajectory.
To illustrate the capabilities of MacSimion 3D, numerical simulations of ion motion in selected regions of a triple quadrupole instrument have been undertaken. The complete instrument is far too complex to handle as a single entity, and must be decomposed into smaller, more manageable problems. Memory and time constraints currently restrict the physical volume of regions which can be studied in MacSimion 3D to about 10 cm3. We are currently exploring the use of phase space methods [1] to compute ion trajectories near the centre of the mass analyzing quadrupoles. We anticipate that future work will focus on a hybrid approach combining numerical calculations with these phase space methods, thereby greatly enhancing the usefulness of such 3D simulations.