Secular frequency

Figure Bl.7.18. (a) Schematic diagram of the trapping cell in an ion cyclotron resonance mass spectrometer excitation plates (E) detector plates (D) trapping plates (T). (b) The magnetron motion due to tire crossing of the magnetic and electric trapping fields is superimposed on the circular cyclotron motion aj taken up by the ions in the magnetic field. Excitation of the cyclotron frequency results in an image current being detected by the detector electrodes which can be Fourier transfonned into a secular frequency related to the m/z ratio of the trapped ion(s).

By expanding the circled region in Fig. 13, the ion-trap stability diagram (Fig. 14) plotted in terms of the parameters az and qz is obtained. These parameters are directly related to the RF (qz) and DC (az) voltages applied to the ion-trap electrodes. The areas of stability have boundaries where the (lu parameters (u — z or r) have values 0 and 1. fju is a complex function of au and qu and is directly related to the fundamental secular frequency of the ion (mu) and the main RF frequency (Q) by the equation... 

Stored waveform inverse Fourier transform (SWIFT) pulses [17] have been applied as a means of broadband ejection of matrix ions generated by Cs+ desorption [18]. These pulses are generated by taking the inverse Fourier transform of the desired frequency domain spectrum and applying the stored time domain waveform to the endcap electrodes via an arbitrary waveform generator. The magnitude of the SWIFT pulse determines the degree of excitation for ions of specific secular frequencies. 

Two other important parameters depending on qz or / will be considered. First, if an RF voltage with an angular velocity co = 2jtv is applied to the trap, the ions will not oscillate at this same fundamental v frequency because of their inertia, which causes them to oscillate at a secular frequency/, lower than v, and decreasing with increasing masses. It should be noted that au and qu, and thus ft, are inversely proportional to the m/z ratio. The relation... 

As the maximum value of ft for a stable trajectory is j> = 1, the maximum secular frequency fz of an ion will be half the fundamental v frequency. We will see later on that this is important for ion excitation or for resonant expulsion. 

If we remember that the secular frequency at which an ion oscillates in the 3D trap is given by... 

Let these ions fragment. Energy is provided by collisions with the helium gas, which is always present. This fragmentation can be improved by excitation of the selected ions by irradiation at their secular frequency. 

In a quadrupole collision cell, the ions undergo multiple collisions. The fragments, as soon as they are formed, are reactivated by collision and can fragment further. In the ion trap, if excitation occurs by irradiation at the secular frequency of the precursor, only this ion is excited, and the product ion may be too cool to fragment further. Figure 2.28 shows an example of this behaviour [17]. 

Operations similar to the 3D traps can be performed, as for example to expel ions of all masses except one and observe the fragmentation, with or without ion excitation at the secular frequency. Then the fragments are analysed. This can be repeated several times for MS" experiments. All the other operations of a 3D trap can be applied, but it also has similar limitations, for example MS/MS is limited to fragmentation scans. Thus precursor ion scan or neutral loss scan that are available with triple quadrupole instruments cannot be used on ion traps (Figure 2.11). 

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. 

What is the behavior of a trapped ion An exhaustive description of the solution of the Mathieu equation and of the operative conditions of an ion trap can be found in the March and Todd books (March and Todd, 1995,2005) on this argument. Just from a pictorial point of view we can say that an ion inside an ion trap follows some periodic fundamental trajectories with well-specified frequencies (called secular frequencies), on which some other periodic motions at higher frequencies (high-order frequencies) are superimposed (Nappi et al., 1997). 

Have secular frequencies inversely related to their m/z values. 

Ions confined by a pure quadrupolar field and within the bounds 0

fundamental frequencies of oscillation in the r and z directions, co, and co , given by Equations 9.9 and 9.10, where n is an integer. The characteristic fundamental secular frequencies (0, 0 and co o) are unique for a given value of P . Equations 9.9 and 9.10 describe the principal frequencies of ion motion in a quadrupolar field. 

The term non-linear resonance describes the resonant absorption of potential energy from higher-order trapping fields. Such absorption occurs when combinations of an ion s secular frequencies match harmonic sidebands of the RF drive frequency. Conditions for non-linear resonance exist in all QITs but the effect on ion motion is dependent upon the order, sign (+ or -) and strength of the superimposed non-linear fields. Due in large part to the work of Franzen and co-workers [87-92], the phenomenon of non-linear fields and their contribution to non-linear resonance effects in ion traps is now well understood. In addition to the quadrupolar resonance, there are a series of resonance conditions desaibed by Equations 9.11 and 9.12 where V is an integer 0 and n is the order of the multipole. The overall order of the resonance (AO is described by Equation 9.13. 

In order to evaluate the changes to the performance of the ion trap, caused by the perturbation to the trapping field due to the addition of holes in the ring electrode to enable our fluorescence measurements, we have carried out ion trajectory calculations in several models of the modified Esquire 3000-1- QIT. Franzen [87,91,92] demonstrated previously the utility of ion trajectory calculations in investigation of the modified hyperbolic angle trap, but considered only ion motion in the axial direction. Here, we discuss results from several SIMION models, which have been constructed with different numbers and sizes of holes in the ring electrode. Fourier analysis of ion trajectories is used to determine secular frequencies, frequency shifts. 

A Summary of the Characteristic Secular Frequencies (o) Expressed in kHz) from Single Ion Trajectories in Q T Models with Progressively More Holes Bored in the Electrodes... 

The theoretical frequencies were calculated using ITSIM v. 5.0 and all ion trajectories were calculated with SIMION, using identical simulation parameters under collision-free conditions. According to Equations 9.9 and 9.10, secular frequencies, co, should be expressed in Rad s" regrettably, it is common practice to express secular frequencies in kilohertz. 

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