Charged Particle in a Quadrupole Field

The basic mathematical equation of the quadrupole mass spectrometer is Mathieu s equation. 

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 solutions to Mathieu s equation comprise an orthogonal set and possess the curious property that the coefficients of their Fourier series expansions are identical in magnitude, with alternating signs, to corresponding coefficients of their Bessel series expansions [2, 3]. Floquefs theorem asserts that any solution of equation (Eq. 20.7) is of the form 

MathieuC and MathieuS are the even and odd Mathieu funetions. These functions may be bounded or not bounded, depending on the parameters a and. All bounded solutions to the Mathieu equation are infinite series of harmonic oscillations whose amplitudes decrease with increasing frequency. 

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