Differential equation Mathieu

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... 

Solving these differential equations which are again of the Mathieu type yields the parameters and... 

The mathematical treatment of these equations of motion uses Mathieu s differential equations. It is demonstrated that there are stable and unstable ion paths. With the stable paths, the distance of the ions from the separation system center line always remains less than r (passage condition). With unstable paths, the distance from the axis will grow until the ion ultimately collides with a rod surface. The ion will be discharged (neutralized), thus becoming unavailable to the detector (blocking condition). 

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 ... 

The motion of an ion injected into the field in the z direction can be described by the Mathieu differential equations ... 

The physical principles underlying the operation of a quadrupole mass spectrometer require the solving of a complicated differential equation, the Mathieu equation. In operation when an ion is subjected to a quadrupoiar RF field, its trajectory can be described qualitatively as a combination of fast and slow oscillatory motions. For descriptive purposes, the fast component will be ignored here and the slow component emphasized, which oscillates about the quadrupoiar axis and resembles the motion of a particle in a fictitious harmonic pseudopotential. The frequency of this oscillation is sometimes called the secular frequency. 

Basically, ions entering the quadrupole move through it in a wavelike trajectory around the z-axis between the quadrupole s electrodes. This motion can be described by the so-called Mathieu differential equations (5.2) and (5.3). 

The theoretical calculation of the surface states is considered by the linear combination atomic orbital (LCAO) method however, in some cases the relation between the electrode potential and the distance from the electrode surface is used. These electric potentials obey the Schrodinger equation and the surface states are obtained as eigenvalues. Inside the crystal lattice, the equation is reduced to the Mathieu differential equation with known solutions. Using the LCAO method, the interaction between the surface states and the electron is avoided. In this case, the proposed wave function is of the form ... 

The potential functions and the resulting energy levels of a one-dimensional hindered rotor, a free rotor, and a harmonic oscillator are compared in figure 6.2. The hindered rotor energy levels can be solved numerically as solutions to the Mathieu differential equation (Abramonitz and Stegun, 1972 Wilson, 1940) and lists of values... 

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... 

Mathieu s equation is a linear second-order differential equation with periodic coefficients. It belongs to the family of ///// s equations. The one-dimensional Mathieu s equation is written in standard form as... 

The French mathematician Mathieu investigated this equation in 1868 to describe the vibrations of an elliptical membrane. Mathieu functions are applicable to a wide variety of physical phenomena, e.g problems involving waveguides, diffraction, amplitude distortion, and vibrations in a medium with modulated density. Hill was interested in the motion of planets and was thus engaged in differential equations with periodic integrals [1]. 

The equation of motion (9.49) is known as Mathieu s differential equation. It has stable solutions only for certain values of the parameters a and b [1221]. Charged particles that enter the trap from outside cannot be trapped. Therefore, the ions have to be produced inside the trap. This is generally achieved by electron-impact ionization of neutral atoms. 

The mathematics of ion trajectory stability within a quadrupole field follows the Mathieu second-order differential equation ... 

Equations of the form [6.13] had been examined in 1868 by the French mathematician Emile Mathieu in connection with vibrations of an elhptically shaped membrane (actually a drumhead ). Solutions to such a differential equation (i.e. expressions describing x or y for an ion as functions of time t) were thus known. These mathematical solutions (Dawson 1976 March 2005) are not derived here. An important feature of the solutions is that they can be divided into two classes, stable and... 

The equations of motion of a single ion inside a linear radio-frequency trap are differential equations of the Mathieu type, which lead to stable or unstable solutions with respect to motion in the jc-y-plane, depending on the trap parameters. A necessary condition for stable trapping of an ion or an ensemble of noninteracting ions is a (Mathieu) stability parameter, q = lQV-gs lm r, of <0.9. Here, Q and m are the ion charge and mass, and ro is the distance from the trap center to the electrodes. 

A graphical representation expressed in terms of reduced coordinates that describes the stability of charged particle motion in a quadrupole mass filter or quadrupole ion trap mass spectrometer, based on an appropriate form of the Mathieu differential equation. 

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