The following section is designed to give an overview of the mode of operation and principles involved in quadrupole ion trap theory, it is not intended as a comprehensive guide to the mathematical theory of ion motion in an ion trap as this is considered to be outside the scope of this section.
Gas phase analytes that are eluted from the gas chromatograph enter the ion trap directly where they undergo electron ionisation. The analyte molecules interact with electrons that are produced from a heated rhenium filament and accelerated across the source at 70 eV. This process leads to the formation of an analyte radical cation. The technique is a “hard” ionization method and the resultant analyte radical cation undergoes extensive fragmentation, so that there is often little to none of the original cation observed. The ionization time in an ion trap is sufficient so that a predetermined number of ions are produced in a process known as automatic gain control. The resultant ions produced in the electron ionization process are gated into the ion trap105.
The quadrupole ion trap consists of three shaped electrodes two of which are virtually identical. These two electrodes are known as the end cap electrodes and have a hyperboloidal geometry. The third electrode is hyperboloidal in shape and is called the ring electrode (Figure 7). The ring electrode is situated symmetrically between the two, end cap electrodes and is isolated from each end cap electrode by means of a quarts spacer106–108.
Figure 7 Cross section of a typical quadrupole ion trap mass analyser illustrating the geometry. F is the filament of the ion source and D is the electron multiplier detector. EC are the end-cap electrode and RE is the centre ring - electrode
As the ions are gated in to the analyser they immediately come under the influence of the trapping potential in the ion trap, this is achieved by the ring electrode having an applied initial RF voltage. This causes ions of a particular m/z to come under the influence of the imposed quadrupole field thus trapping them. The motion of the ions within the quadrupole field can be described mathematically by the Mathieu equation105.
𝑑𝑑2𝑢𝑢
𝑑𝑑2𝜉𝜉 +(𝑐𝑐𝑢𝑢− 2𝑞𝑞𝑢𝑢𝑐𝑐𝑐𝑐𝑠𝑠2𝜉𝜉)𝑢𝑢 = 0
Equation 14
Where: u represents the coordinates of x, y and z. a and q are the trapping parameters and are dimensionless. 𝜉𝜉 is defined as follows:-
𝜉𝜉 =𝜔𝜔𝑐𝑐2 = 2𝜋𝜋𝜋𝜋𝑐𝑐2
Equation 15
Where
f is the fundamental RF frequency of the ion trap (ca. 1MHz)
𝜔𝜔 is angular frequency applied to the RE electrode
t is time
In a conventional quadrupole mass analyser the ion motion that results from the applied potentials to the 4 rods of the quadrupole occurs in two directions (x and y) with the motion on the z axis accruing from the acceleration of the ions from the ion source, optics as they enter the quadrupole. With the ion trap however the motion of the ions in the quadrupole field occur in three dimensions x, y and z. When the geometry of the ion trap is taken in to account the resulting cylindrical symmetry results in X2+Y2 = r2 and so the motion of ions in an ion trap may be expressed in terms of r and z coordinates106.
In terms of the Mathieu equation if u can be either r or z then as a result the coefficients a and q may be summarised using Equations 16 and 17
𝑐𝑐𝑢𝑢 = −2𝑐𝑐𝑟𝑟= 𝑚𝑚(𝑟𝑟−16z𝐴𝐴𝑒𝑒 02+ 2𝑧𝑧02)𝜔𝜔2 Equation 16 𝑞𝑞𝑢𝑢 = −2𝑞𝑞𝑟𝑟 = 𝑚𝑚(𝑟𝑟 8z𝐴𝐴𝑒𝑒 02 + 2𝑧𝑧02)𝜔𝜔2 Equation 17
Where m = mass of an ion e = electronic charge
z = number of charges on the ion V = is the RF frequency or AC voltage
U = the DC potential and is proportional to au and is equal to zero
For an ion to have a stable trajectory within the mass analyser, the ions must have simultaneous stability in both the r and z axes. By plotting qu against au it is possible to illustrate where the regions of stability residue. This type of plot is known as a Mathieu stability diagram (Figure 8). Those regions which overlap closest to the origin provide the stable trajectory necessary for an ion to remain caged in the quadrupole field. The stability of the ion trajectory is dependent on the stability parameters βr and βz which are in turn dependent on a and q. The limits of this first stability region are defined by 0< βr ,βz <1
Figure 8 Mathieu stability diagram, the shaded indicates the regions of a/q space that overlap and provide stable trajectories for an ion to remain in the quadrupole field.
As U=0 (i.e. no DC voltage is applied) the ion trap will operate along the qu axis. At a
given U/V ratio ions of varying m/z can be found to reside in a line that crosses the stability region. Ions with the highest m/z are located closer to the origin than those with lower m/z ratios. The regions of stability found in a/q space can then be plotted as envelopes that have a characteristic shape. If V is increased qz is also increased for each ion. At the point where qz = 0.908 (az = 0), β = 1 and the ion has reached its stability limit. If V increases past this point the ion’s trajectory becomes unstable and the ion is then ejected from the ion trap in the z axis. This mode of operation is known as mass-selective instability mode and as the RF or V is continually increased ions of increasing m/z are ejected from the mass analyser and detected resulting in a full scan mode spectrum106
Figure 9 Example of ion ejection at the stability limit.