Mathieu parameters

The ion trap works on a similar principle to the linear quadrupole. Oscillating RF and DC voltages are applied to the electrodes. These voltages create a quadrupolar field in the ion trap, and ions can be retained in stable trajectories in this field. The force acting upon the ions in the trap is directly proportional to the distance of the ions from the center of the ion trap. Therefore the quadrupolar field acts to store the ions as a packet at the center of the trap. For an ion to be retained in the ion trap it needs to have a stable trajectory in both the axial (z) and radial (r) directions. Whether the ion is stable in one or both of these directions is given by the solutions to the Mathieu equation previously used to describe ion motion through the linear quadrupole with the reduced Mathieu parameters ... 

Usually no DC voltage is applied (az — 0) and the RF frequency and trap geometry are fixed so the expression for the reduced Mathieu parameter that determines stability has the form... 

However, the Mathieu parameters a and q, defined by Eqs. 2.9 and 2.10, can still be employed for the description of the DIT theoretical stability diagram, considering the U and V values as the average values of the dc and ac components of the rectangular wave voltage applied to the intermediate electrode. They are defined as... 

Quadrupole Mass Filters Quadrupole mass analyzers consist of four electrodes, ideally of hyperbolic rods, that are accurately positioned in a radial array. For practical as well as economic reasons, most quadrupole mass filters have employed electrodes of circular cross section. A potential is applied to one pair of diagonally opposite rods consisting of a DC voltage and an rf voltage. To the other pair of rods, a DC voltage of opposite polarity and an rf voltage with a 180° phase shift are applied. The ion motion under the influence of this two-dimensional (2D) field can be described mathematically by the solutions to the second-order linear differential equation, known as Mathieu equation, from which the Mathieu parameters, and can be derived as... 

Example 3.4 Suppose that a quadrupole mass filter is operated at the tip of the first stability region, where the Mathieu parameters a and qu are equal to 0.237 and 0.706, respectively. If the maximum rf potential is 6000 V, the rf frequency is 0.5 MHz, and ro is 1.0 cm, what maximum miz value can be analyzed with this instrument ... 

Commercial ion traps are not of ideal geometry, but are stretched, for which the expression = 2z no longer holds. The values of the Mathieu parameters for such traps are obtained by replacing with Tq - - 2z in Eqs. (3.15) and (3.16). The plot of a and q gives several Mathieu stability regions (shown in Figure 3.15). 

Because fluorescence detection by repeated absorption-emission cycles is not applicable to trapped molecular ions in UHV, that is, in the absence of collisions with a buffer gas [68], different techniques are required for their reliable identification. A commonly used destructive technique for molecular ions is time-of-flight (TOP) mass spectroscopy. We have used a simplified variant in the Ba+ apparatus. The trapped ions are extracted from the trap by reducing the radio-frequency amplitude, in the presence of a finite dc quadrupole potential Vo, which causes the ion trajectories to become unstable (the Mathieu -parameter enters the instability region). Heavy and hot ions escape first. Upon leaving the trap, the ions are guided to and attracted by the cathode of a channel electron multiplier (CEM) and counted. 

In practice, there is no DC conponent, and thus, only the Mathieu parameter q is required, which is obtained by solving the above equations... 

One specific, but often cited geometry of the ion trap is defined by Eq. (9.5). Solving for Zq in Eq. (9.5) and substituting the result into Eqs. (9.3) and (9.4) above, simplifies these Mathieu parameter equations. 

Plotting and overlapping the solutions to the Mathieu equation in a, q) space for the r and z dimensions forms the QIT stability diagram. A portion of the stability diagram including the solutions where = 0 is shown in Figure 9.3b. The Mathieu parameters, a and q, are indicative of the stability and motion of an ion in the 3D-quadrupole field. 

Figure 8.10 Stability diagram for an infinitely long and perfect quadrupole in the dimensionless space of (a, q), where a and q are the Mathieu parameters given by equations (8.2.4) in the text. Ions that are within the enclosed region have finite (stable) trajectories, independent of their initial displacement and initial phase of the applied r.f. Ions that are outside this region have amplitude of their oscillations exponentially increasing and are thus unstable in the field - independent of their initial displacement and initial phase of the applied r.f. Also shown are the iso-beta lines ions of the same beta have same frequencies of trajectory oscillation in the field.

In most practical cases, the sequential chemistry involves primary ions (either plasma-based matrix ions or primary in-cell produced product ions) that are significantly different in m jz compared to the analyte ion of interest. Our example of sequential (in time) chemistry with CH4 can be presented on the mass scale, as shown in Figure 8.20. Let us assume that the instrument is operated in such a manner that when the analyzing quadrupole downstream of the cell is detecting Ca+ ions, the Mathieu parameters of the quadrupole field of the cell are set... 

Figure 8.20 Sequential chemistry with methane (solid lines) and impurities (dotted lines) plotted along the m/z scale. When m/z = 40 is transmitted through the quadrupole field at Mathieu parameters a = 0.1, q = 0.5, the ions of m/z < 26 or m/z > 49 (inside shaded area) are unstable in the quadrupole field,...

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