Fundamental secular frequency

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... 

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. 

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. 

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). 

Ions of a specific miz in the 2D quadrupole field have fundamental frequencies of motion unique to their values, and these secular frequencies can be used to resonate these... 

The problem of expressing, for the H20 molecule, the relationship between the frequencies of the fundamental modes and a set of force constants has now been solved in such a way that the equations are as simple as symmetry will permit them to be. In this case, the secular equation is factored into one of second order for the two A, vibrations and one of first order for the sole Bt vibration. The explicit forms of these separate equations are... 

Properties. It is of considerable importance to examine the nature of die solutions obtained above. It is evident from Eq. (9), Sec. 2-2, that each atom is oscillating about its equilibrium position with a simple harmonic motion of amplitude Aik — Kkhk, frequency x /27t, and phase e. Ihirthermore, corresponding to a given solution X of the secular equation, i he frequency and phase of the motion of each coordinate is the same, but I lie amplitudes may be, and usually are, different for each coordinate. On account of the equality of phase and frequency, each atom reaches its position of maximum displacement at the same time, and each atom pa.sscs through its equilibrium position at the same time. A mode of ibration having all these characteristics is called a normal mode of vibra-iion, and its frequency is known as a normal, or fundamental, frequency of (he molecule. 

A three-dimensional representation of an ion trajectory, shown in Figure 8, has the general appearance of a Lissajous curve composed of two fundamental frequency components, rUf O and >z,o of the secular motion. Higher-order (n) frequencies exist and the family of frequencies is described by given by... 

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