Ion energy in quadrupole mass spectrometry

Abstract

Application of an analytical solution of the Mathieu equation in conjunction with algebraic presentation of the Mathieu functions for description of the ion energy in a radiofrequency quadrupole field is discussed in this work. The analytical approach is used to express the ion energy averaged over the initial ion velocity distribution function, field phase and ion residence time. Comparisons with the approximate solutions for potential ion energy are presented with demonstration of their limits. Application of the method for different stability regions is discussed. Mathieu functions [1][2][3] has simplified theoretical description of the motion of ions in a quadrupole field. As a result of this progress, the algebraic aspects of the Mathieu functions were implemented "simply as another special function". The analytical method of the solution of the Mathieu equation in conjunction with algebraic presentation of the Mathieu functions allows introduction of simplifications and generalizations, delivering an alternative method to both the numerical solution of the Mathieu equation and to the matrix method. Previous attempts to use the Mathieu functions were too general due to the absence of simple algorithms or too local due to the necessity to use expansion series around small quantities (see [4] and references therein). The analytical method, in contrast to the matrix method, utilizes a single solution for a complete ion trajectory. The closed formulae obtained provide a general and preferable method for analytical expression of the fundamental properties of the quadrupole field such as ion trajectory stability, transmission/acceptance, resonance (see [5]), and momentum/energy characteristics of the ion motion. The linear quadrupole and the quadrupole trap are considered in this work as examples that demonstrate the advantages of the analytical method for the determination of the fundamental properties of the mass selecting devices as well as their effect on the ion energy. Illustrations of practical implementation of the method are also given for different regions of stability. In this work it is demonstrated that in order to evaluate the average ion energy in the quadrupole RF field, one has to calculate two dimensionless parameters 21 2 ͑a, q͒ and 22 2 ͑a, q͒, which depend only on the field properties. Linear QuadrupoleFor the two-dimensional quadrupole capacitor, the potential of the RF driven quadrupole field can be expressed as a combination of two terms having spatial (U) and periodic (V) trapping potentials:where r 0 is the electrode separation, is the main trapping RF angular frequency and is the initial phase of the main trapping frequency. In the vicinity of the z-axis, the electrostatic field of such a quadrupole has the equipotential lines forming a hyperbolic surface. That leads us to the system of equations for motion of an ion (with mass m and charge e) having arbitrary initial conditions {x i , y i , z i } and {ẋ i , ẏ i , ż i } (i-initial, t i ϭ 0) in Cartesian system of coordinates:Ά mẍ ͑t͒ ϭ ...

Intramolecular hydrogen bonds and charge‐remote fragmentations in negative‐ion mass spectrometry of dihydroxylated fatty acids

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“…The reaction mechanism for the 2Me-thiophene was not further explored since we were interested only in explain the low yield of reaction 2 in this case. The energy necessary to overcome the barrier to the insertion product can derive from a contribution of the radio frequency field to the ion energy [30][31][32].…”

Development of the analytical method of the general mathieu equation solution

“…-on motion stability and resonance phenomena in quadrupole traps and filters used in mass spectrometers [10,[12][13][14][15][16][17][18][19][20][21][22][23].…”