The families of solution boundaries for the Mathieu equation that lie near the origin are depicted in Figure 2, showing four distinct regions of stable trajectories for ions passing through the quadrupole, utilizing the Mathieu a and q parameters. Graphical treatment can answer the questions such as ‘Does the ion have a stable trajectory at the voltages applied? Will the ion pass through the quadrupole?. This represents that in each of the x and y direction, the motion is sinusoidal, comprising micromotion at the harmonic frequencies added in and macromotion at the fundamental frequency (ξ 0). ![]() Considering ion trajectories as infinite sums of sine and cosine functions, with each succeeding term comprising higher frequency and smaller amplitude, is acceptable. The aforementioned equation is intuitively reduced to a similar infinite sum of sine and cosine functions. The solution to this second order linear differential equation is as follows: In the above equations, u= Position along the coordinate axes (x or y) and is represented as Ωt/2, in which t is time and Ω is the applied RF frequency e = the charge on an electron U = Applied DC voltage V = Applied zero-to-peak RF voltage m = Mass of the ion and r 0= The effective radius between electrodes. When discussing quadrupole theory, it is customary to mention the Mathieu Equation: Schematic of a typical quadrupole power supply connections is illustrated in Figure 1. Two electrical connections are required for the quadrupole. Striking the quadrupole electrodes neutralizes the ions with unstable trajectories. For a specific system, the amplitude of the voltages identifies the mass or range of masses that have stable trajectories through the quadrupole. The application of a combination of precise DC and RF voltages to the quadrupole rods helps focusing the ions to be mass analyzed down the quadrupole center. Schematic of typical quadrupole power supply connections The common quadrupole fabrication technique involves positioning the four round poles in such a fashion that the centers of the poles coincide with the corners of an imaginary square.įigure 1. The orientation of the electrodes is such that there is a hyperbolic (quadrupolar) electric field between them. Are there any areas in either drawing in which there are no field lines? d.A quadrupole mass filter is composed of four mutually parallel, highly precise, electrically-isolated electrodes. Do field vectors point in different directions anywhere in the drawings? c. Do field lines start or end in different locations? b. Your discussion must at least include the following aspects: a. Write a paragraph discussing the differences between the two situations. The drawings you made should be different in some respects. The directions of the field vectors must be indicated (using arrow tips). Field lines in the inner region and in the outer region of the quadrupole (separated by the dashed lines). Draw the electric field lines for both situations into the two figures. In B), a metal sphere is located in the center of the charge arrangement. ![]() Figure 2B has an additional element, a metal sphere in the center of the quadrupole. Activity 2: The electric quadrupole, revisited Figure 2A shows the electric quadrupole from last week again.
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