Mathieu stability diagram

FIGURE 17.7. Mathieu stability diagram. Figure from March and Hughes (1989), pg. 50. 

The Mathieu stability diagram plots o-values versus qi-values and marks the boundaries where ions will be confined to the trap. Figure 2.13 is a plot of the Mathieu stability diagram with four points A-D marked. The trajectory of ions at those four points is shown in plots A-D on the right. Point A is within the r and z stability regions of the diagram, as shown in plot A, while point B is stable in the z-direction but unstable in the r-direction and falls out of the trap radially. Point C is stable in the r-direction but unstable in the z-direction and falls out... 

Figure 2.13 Mathieu stability diagram with four stability points marked. Typical corresponding ion trajectories are shown at the right. Please observe that diagrams A-C are rotated 90 degrees relative to Figure 2.12.

The Ion Trap mass filter is closely related to Quadrupole mass filters in that the action of both is described using Mathieu stability diagrams (March 1997). This is realized in that a Quadrupole mass filter applies a two-dimensional time-dependent... 

Mathieu stability diagram A diagram describing the operation of Quadrupole and ion trap mass filters... 

FIGURE 7.5 Simplified Mathieu stability diagram of a quadrupole mass filter, showing separation of two different masses A and B. (From Quadrupole Mass Spectrometry and Its Applications P. H. Dawson, Ed., Elsevier, Amsterdam, 1976 reissued by AIP Press, Woodbury, NY, 1995.)... 

Figures The Mathieu stability diagram in space for the quadrupole ion trap in both the r and z directions. Regions of simultaneous overlap are labelled A and B.

Figure 2.3 Mathieu stability diagram. The ion motion is stable if both operating points lie within the bounded region. The plotted points are for (qx = 0.7, ax = 0.1) and (qy = —0.7, Oy = —0.1)...

The Mathieu equation for the quadnipole ion trap again has stable, bounded solutions conesponding to stable, bounded trajectories inside the trap. The stability diagram for the ion trap is quite complex, but a subsection of the diagram, correspondmg to stable trajectories near the physical centre of the trap, is shown in figure Bl.7.15. The interpretation of the diagram is similar to that for tire quadnipole mass filter. 

Fig. 11.9. The stability diagram for a quadrupole mass filter. The shaded area represents solutions for the Mathieu equations that result in stable ion trajectories through the device. The area outside the boundaries represents solutions for the equations that produce unstable trajectories.

Resolution versus Sensitivity. A quadrupole mass filter can be programmed to move through a series of RF and dc combinations. The Mathieu equation, which is used in higher mathematics, can be used to predict what parameters are necessary for ions to be stable in a quadrupole field. The Mathieu equations are solved for the acceleration of the ions in the X, Y, and Z planes. A selected mass is proportional to (dc x RF x inner radius)/(RF frequency). For a given internal quadrupole radius and radio frequency, a plot can be made of RF and dc values that predict when a given mass will be stable in a quadrupole field. This is called a stability diagram (Figure 13.3). RF and dc combinations follow the value shown... 

Stability diagram along r and 2 respectively for a 3D ion trap. The iso/S lines for / = 0 (solid lines) and fSu = 1 (dotted lines) are drawn. The areas inside these limits correspond to stable trajectories for the considered coordinate. They correspond to imaginary solutions of the Mathieu equation. 

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... 

The stability diagram of the Mathieu equation in the plane of the parameters S and is called the Strutt-Ince diagram (van der Pol and Strutt 1928). This diagram can be transformed to the plane of the forcing amplitude r and the forcing frequency /= col2%. 

Exactly as for the hnear quadrupole, the Mathieu equations for a Paul trap provide solutions to the ion s equations of motion that are of the stable and unstable types. The details of derivation of these solutions (March 2005) are omitted here. The stabihty and instability solutions are again conveniently represented on a stability diagram in which the stabihty regions of interest are those in which the motions in both the z- and r-directions are stable. The double stabihty region that is exploited in almost aU traps is shown in Figme 6.18, plotted in (a, qf) space for convenience (the radial diagram is inverted and compressed by a factor of two since the corresponding stabihty factors differ by a factor of —2, see above). 

Fig. 3.2 Field generating potentials a and stability diagram (Mathieu diagram) b of a QMS. The blue shaded areas represent values of a and q which give rise to bounded trajectories. The complete blue area defines the stability region for ions when passings through the QMS for RF-only conditions (a = 0), however when applying an additional DC potential (in the plotted case the ratio a/q = 2U/V = const.) only the dark blue area is stable, defined by the orange line—also known as the mass scan line [25]...

The solutions to Mathieu s equation are of two types, (i) periodic but unstable, and (ii) periodic and stable. Solutions of type (i) form the boundaries of unstable regions on the stability diagram and correspond to those values of a trapping parameter, j z that are integers, that is, 0, 1, 2, 3,. / z is a complex function of az and qz that is... 

Figure 5 Several Mathieu stability regions for the three-dimensional quadrupole field. (A) Diagrams for the z direction of space. (B) Diagrams for the rdirection of space.

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.

We now consider the graphic solutions of the Mathieu equations. Figure 4.6 shows the shapes of the stability diagrams of the ions in the x and y directions (within the orthonormed system shown in Figure 4.5) according to the parameters a and q. 

Hiroki, S., Abe, T, and Murakami, Y. (1992) Separation of helium and deuterium peaks with a quadrupole mass spectrometer by using the second stability zone in the Mathieu diagram. Rev. Sci. Instrum., 63(8), 3874-6. 

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