A Paul trap or quadrupole ion trap is a device designed to confine ions or charged particles in a given
space. It consists of four electrodes that produce electric fields varying in time. The voltage applied to these
electrodes varies harmonically with low frequency (quasi-static regime), which simplifies the model. The
goal of this project is to numerically simulate in matlab a Paul trap and the motion of the ions trapped in it
by discretizing the Poisson equation and applying the method of moments (MoM), first in 2D and then
generalized to 3D2019/202

Mercader, Mikel

Sanz, Axel

English

UPCommons. Portal del coneixement obert de la UPC

M​ODELLING​ ​AND​ ​DESIGNING​ ​A​ P​AUL​ ​ION​ ​TRAP Modelling and designing a Paul ion trap    Axel Sanz and Mikel Mercader Universitat Politècnica de Catalunya (UPC) June 17, 2020 A Paul trap or quadrupole ion trap is a device designed to confine ions or charged particles in a given                    space. It consists of four electrodes that produce electric fields varying in time. The voltage applied to these                  electrodes varies harmonically with low frequency (quasi-static regime), which simplifies the model. The             goal of this project is to numerically simulate in matlab a Paul trap and the motion of the ions trapped in it                      by discretizing the Poisson equation and applying the method of moments (MoM), first in 2D and then                 generalized to 3D.  I. M​ATHEMATICAL​ ​BASIS​, M​O​M The first step is to formulate the electromagnetic        equations that rule our system. These are integral equations,         that we can discretize using the MoM. This method         transforms our equations into a linear system which we can          solve to find the potential at every point in space. We start with Poisson’s equation   And Green’s function that satisfies, in the 3D case  And so we obtain an expression for the potential:   If we impose that the potential at a given surface must be            a certain value (boundary conditions) we arrive to the         following integral equation.   In the 2D case, Green’s function is  And the integral equation we obtain is    For our simulations we need to discretize the charged         surfaces. To do this, we will create a triangles mesh with           constant charge in the center of every triangle. Using the          function meshgrid we can create a 2-dimensional grid, and         then convert it into a 3-dimensional one by computing the z           coordinate in function of the x and y ones. By this method            we can obtain a vertex matrix, and then a topology matrix           by using the delaunay function. Now the method of moments allows us to transform the          integral equation into a linear system. The unknown charge         is now discretized and can be expressed as a linear          combination of basis functions. The new unknowns will be         the coefficients of this combination.  The potential then can be approximated by applying L, a          linear operator, to this charge:  To make this approximation as accurate as possible, we         set to zero the inner products of the error by introducing a            set of weighting functions Wm(r). By imposing       We obtain the following linear system of equations[​1​]:   Solving this system we obtain the coefficients q and thus          the charge and the potential. M​ODELLING​ ​AND​ ​DESIGNING​ ​A​ P​AUL​ ​ION​ ​TRAP II. S​IMULATION​ ​IN​ 2D We simulated the potential that two circular plates would         produce, one at 5V and the other at -5V, separated by a            distance of 4 units and with a radius of 1. With this example             we tested how the MoM can be used to calculate the           potential created by a set of objects. We represented the          potential obtained in the following figures:    figure 1​. ​Potential due to the circular plates. The simulation is consistent with the expected result. We          also computed the capacitance numerically, with the       obtained discrete charges, and confirmed that it is almost         equal to the theoretical capacitance (with a relative error of          ~0.1%).     III. T​HE​ ​ION​ ​TRAP Usually, Paul Ion Traps are designed using       hyperbolically shaped poles instead of plane plates, spheres        or any other different simpler surfaces. The reason behind         this is that hyperboloid-like poles generate almost constant        Electric Field at their centers, making the confinement of         ions much easier. In fact, hyperboloids are the optimal         shape for the design of the poles (it can be analytically           proven but it is not the aim of the project).   figure 2​. ​Poles distribution of the Ion Trap. In our case, we used the design shown in figure 1. The            vertical poles in red are situated at z​0​=+2.5 and z​0​=-2.5, the           blue pole has a minimum radius from the center r​0​=5. With           this information we can obtain their corresponding       equations, which are used to discretise the surfaces:  q1.        z   E =  ± √( ) .5r2z0 2 + 0 2 (Vertical)  q2.        r   E = √ r z02 + 2 2 (Horizontal)  To keep the ions confined in the center, a harmonic          potential of the form is applied    (t) · in(2πf )V = √2 V 0 · s · t    at the poles. We can change the values of V​0 ​and           f  according to the type of ions we need to trap. In general, we             would want to use a relatively low frequency in order to           consider quasi-static regime. However it has to be taken         into account that the lower the frequency is the more likely           the ions are to escape, since we are headed to constant           voltage. Another important factor to consider is that the         applied voltage V​0 must be small enough because if the          electric field is strong the particles are pulled to the poles           faster than the time it takes to change the polarity.  M​ODELLING​ ​AND​ ​DESIGNING​ ​A​ P​AUL​ ​ION​ ​TRAP   figure 3​. ​Potential at y=0 plane for V(T/4)= .√2  For particles with a charge/mass ratio (R​qm​) of 1, as          protons, we found stability when using V​0 ​= 10 V and           . Stability is preserved if lower voltage or500 Hzf = 1         higher frequencies are used; on the contrary, decreasing the         frequency and raising the potential causes the ions to leave          the trap.  It is possible to design a trap for any type of R​qm only             varying the potential, this is due to the fact that:  Ea = mF = mqE = qm = Rqm (− V )∇   The potential in every point in space V is directly          proportional to the voltage applied to the poles V(t) and          consequently to V​0​. If we found stability for R​qm​= 1 and           V​0 ​= 10 V, we need to keep constant the product R​qm​V​0​=10            obtaining the final relation:   V 0 =10Rqm  IV. S​IMULATION​ ​RESULTS With the parameters of the trap already chosen, we can run a simulation of the movement of the ions, placing them near the center with zero initial velocity:  figure 4​. ​x, y and z coordinates of confined ions over time [ms]  figure 5​. ​Path described by the confined ions     As we see in both figure 4 and figure 5, when using the proper voltage and frequency the ions remain near the center of the trap following periodic movements. However, if the parameters of the trap are wrongly designed, such as an excessive potential or low frequency, we observe the following: M​ODELLING​ ​AND​ ​DESIGNING​ ​A​ P​AUL​ ​ION​ ​TRAP  figure 4​. ​x, y and z coordinates of not confined ions over time [ms]  V. C​ONCLUSIONS With this project we show how one can use programming and numerical methods to understand and work on electromagnetic problems. We have successfully simulated a Paul ion trap, allowing us to freely experiment with the parameters to seek the optimal ones for a real ion trap. VI. R​EFERENCES  [​1​] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [​2​]  Sadasiva M.Rao, Allen W.Glisson, Donald R.Wilton and B. Sarma Vidula, “A Simple Numerical Solution Procedure for Statics Problems Involving Arbitrary-Shaped Surfaces“.    figure 5​. ​Path described by the not confined ions     As seen in both figures 4 and 5, the ions escape the trap and are headed to the poles. In this precise case the used parameters were V​0 ​= 12V and f =1400 Hz.            

Modelling and designing a Paul ion trap

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Modelling and designing a Paul ion trap

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