Mathieu equations

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. 

This shows that p (energy) increases exponentially at a fixed value of the phase series solution in the form... 

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.

In case of an inhomogenous periodic field such as the above quadrupole field, there is a small average force which is always in the direction of the lower field. The electric field is zero along the dotted lines in Fig. 4.31, i.e., along the asymptotes in case of the hyperbolic electrodes. It is therefore possible that an ion may traverse the quadrupole without hitting the rods, provided its motion around the z-axis is stable with limited amplitudes in the xy-plane. Such conditions can be derived from the theory of the Mathieu equations, as this type of differential equations is called. Writing Eq. 4.24 dimensionless yields... 

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... 

The force increases linearly from zero of the centre of the quadrupoie. The force in the x direction is independent of the position, that means the x and motions are independent and can be considered separately. The ion motion in a quadrupoie can be described in the form of the Mathieu equation. Substituting Equation (3.16) in to Equation (3.19) and considering that the acceleration of ions in the x direction is given by ax = d2 x/dl2 then the differential equation of ion motion results in ... 

Mass spectrometer that consists of four parallel rods whose opposing poles are connected. The voltage applied to the rods is a superposition of a static potential and a sinusoidal radio frequency potential. The motion of an ion in the x and dimensions is described by the Mathieu equation whose solutions show that ions in a particular m/z range can be transmitted along the z-axis.1... 

Application to Mathieu functions. As an application of (5. 3) consider the Mathieu equation... 

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

To calculate trajectories, the forces F, Fy and Fz exerted on the ion must be known (Mathieu equations). Because this operation involves complex computation, only two particular cases will be discussed. 

To allow the Mathieu equation to be used, the factor is introduced, where... 

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 ... 

Flere again, a mathematical analysis using the Mathieu equations allows us to locate areas in which ions of given masses have a stable trajectory. These areas may be displayed in... 

In using the Mathieu equations to locate areas where ions of given masses have a stable trajectory, the equations are very similar to the ones used for the quadrupole. However, in the quadrupole, ion motion resulting from the potentials applied to the rods occurs in two dimensions, x and y, the z motion resulting from the kinetic energy of the ions when they enter the quadrupole field. In the Paul ion trap the motion of the ions under the influence of the applied potentials occurs in three dimensions, x, y and z. However, due to the cylindrical symmetry x2+y2 = r2, it can also be expressed using z, r coordinates (Figure 2.13). 

In these equations, z is used for the number of charges, to avoid confusion with the z coordinate. The general form of the Mathieu equation, whose solutions are known, is... 

To have a stable trajectory, the movement of the ions must be such that during this time the coordinates never reach or exceed r0 (r-stable) and z0 (z-stable). The complete integration of the Mathieu equation by the method of Floquet and Fourier requires the use of a function e( +I. Real solutions correspond to a continuously increasing, and thus unstable, trajectory. Only purely imaginary solutions correspond to stable trajectories. This requires both a = 0 and 0 ftu < 1 (Figure 2.14). 

This pu parameter can be calculated from the q, and au parameters of the Mathieu equation. Exact values require the use of long series of terms. The following equation allows approximate values to be obtained [12] ... 

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. 

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