Hyperbolic electrode

In case of an inhomogenous periodic field such as the above quadrupole field, there is a small average force which is always in the direction of the lower field. The electric field is zero along the dotted lines in Fig. 4.31, i.e., along the asymptotes in case of the hyperbolic electrodes. It is therefore possible that an ion may traverse the quadrupole without hitting the rods, provided its motion around the z-axis is stable with limited amplitudes in the xy-plane. Such conditions can be derived from the theory of the Mathieu equations, as this type of differential equations is called. Writing Eq. 4.24 dimensionless yields... 

The ion trap is a similar analyzer. There are two end cap electrodes that are at ground potential. An electrostatic field is generated hy a donut-shaped hyperbolic electrode within the cap, which maintains ions in a stable trajectory. Changing electrode voltages ejects ions of a particular mass from the trap into the detector (Honour, 2003). 

Improved separation behaviour is observed in quadrupole analyzers with hyperbolic rods (see Figure 3.9). The quadrupole field is produced by four parallel hyperbolic electrodes, whereby Equations 3.16-3.17 can be applied. 

Figure 16.9—Representation of a quadrupole. Notice the pairing of oppositely charged electrodes. This experimental design requires high-precision machining of the hyperbolic electrodes. To the right a series of equipotential hyperbolic lines in the central part of the quadrupole is shown.

The operating principle of a quadrupole ion trap mass analyzer is similar to that of a standard quadmople but the geometry is different. A trap consists of three hyperbolic electrodes comprising a ring and two endcap electrodes. Applying voltages to these electrodes, results... 

Ion trap, by which ions of a given mass are trapped within a set of three hyperbolic electrodes with a DC electric held on each of two end-cap electrodes and an oscillating electric field on the ring electrode. Ions are injected in a pulsed manner, stored (or trapped) for a short amount of time, then extracted one mass at a time by the pulsed extraction grid. These instruments can detect molecules up to molecular 70,000 Da with very high sensitivity (Figure 1.8). 

The ion trap is a variant of the quadrupole mass spectrometer, which was first described by Paul and Steinwedel in 1960. It consists of a circular hyperbolic electrode, with spherical electrode caps at the top and bottom, and can be conceived as a quadrupole bent round in a circle. The three electrodes correspond to three of the rods of the bent quadrupole, while the fourth (inner) rod is reduced to a vanishingly small point. 

The cylindrical device is easier to construct than a 3D ion trap having hyperbolic electrodes, and variation of the ratio rc 2zc (radius of cylinder to height of CIT) is facile. Research into QTs has focused upon small CITs, described as mini-CITs, for application as tandem mass spectrometers and for the development of arrays of mini-CITs. In order to achieve in a mini-CIT a given value of z(= qx from eqn [2]), an elevated drive frequency is required to compensate for the reduced value of r. In this manner, a potential well of sufficient magnitude is facilitated that permits excitation of ions confined within the mini-CIT. 

The QIT consists of two hyperbolic electrodes serving as end caps along with a ring electrode that replaces two of the linear quadmpole rods, i.e., it could theoretically be obtained by rotating a linear quadmpole with hyperbolic rods by 360° (Fig. 4.43, 4.44). Thus, a section through the rz-plane of the QIT closely resem-... 

Fig. 4.59. Important ICR cell types electrodes are labeled with E excitation, D detection, T trapping (a) classic cubic ceU, (b) Penning cell with hyperbolic electrodes, (c) infinity Cell with segmented trapping plates, and (d) triple cell. The latter two aim at virtually expanding the potentials in the trapping zone along the z-axis. Adapted from Ref. [197] with permission. Elsevier Science Publishers, 1995.

The Penning trap consists of a homogeneous magnetic field in the z direction and a dc electric field formed by hyperbolic electrodes resulting in a dc electric potential... 

In 2004, the Cooks group at Purdue built a rectilinear-LQIT (RLQIT) with rectangular electrodes, and they previously built cylindrical ion traps (CIT). Both of these traps are attractive alternatives to XQITs that use hyperbolic electrodes because they are easier to build. 

Figure 13.10 shows the calibration curve of the LED-based optical oxygen sensor compared with the calibration curve of a commercially available Clark-type sensor (Ingold Electrodes, Wilmington, Massachusetts). While the Clark-type shows a linear calibration, the optical sensor allows a hyperbolic response as predicted by the Stem-Volmer-type equation 72 ... 

A linear quadrupole mass analyzer consists of four hyperbolically or cyclindrically shaped rod electrodes extending in the z-direction and mounted in a square configuration (xy-plane, Figs. 4.31, 4.32). The pairs of opposite rods are each held at the same potential which is composed of a DC and an AC component. 

Theoretically, each electrode should have a hyperbolic cross section for optimized geometry of the resulting quadrupole field, and thus for optimized performance. [103,104] However, cyclindrical rods are often employed instead, for ease of manufacture. By adjusting the radius of the rods carefully (r = 1.1468ro), a hyperbolic field may be approximated. [113] However, even slight distortions of the ideal quadrupole field either from interference with external fields or due to low mechanical precision or inadequate shape of the device cause severe losses of transmission and resolution. [114] The expected advantages of hyperbolic rods [115] have been demonstrated by ion trajectory calculations [110,116] circular rods cause a reduction in macromotion frequency because of an increased residence time of the ions in close vicinity to the rods this in turn means reduced resolution. 

For the QIT, the electric field has to be considered in three dimensions. Let the potential ring electrode (jcy-plane) while -cylindrical coordinates by the expression [137,141]... 

Gibson, J.R. Taylor, S. Prediction of Quadrupole Mass Filter Performance for Hyperbolic and Circular Cross Section Electrodes. Rapid Commun. Mass Spectrom. 2000,14, 1669-1673. 

Figure 3.9 Quadrupole mass analyzer with hyperbolic-shaped rods. A potential of + 0 is applied to the electrodes in the x direction and — 0 to the electrodes in the direction. The potential at the centre is zero. Equipotential contours have a hyperbolic shape.

An ideal quadrupole field can be generated using four parallel electrodes (Z, = 5 to 20 cm) which have a hyperbolic cross-sectional field at their interior (Fig. 16.9). The electrodes are coupled in pairs and a potential difference U is applied across the pairs. If the distance between two opposite electrodes is 2 r0, then the potential d> within the xy plane of the quadrupole will be given by ... 

A recent effort of this laboratory has resulted in the introduction of a hyperbolic Penning trap (3 p. The cell consists of two end caps and one ring electrode similar to the design of Byrne and Farago (51.) (see Figure 8). 

The ion trap generally consists of three electrodes, one ring electrode with a hyperbolic inner surface and two hyperbolic endcap electrodes at either end (a cross section of an ion trap is found in Figure 1.11). The ring electrode is operated with a sinusoidal radio frequency field while the endcap electrodes are operated in one of three modes. The endcap may be operated at ground potential, or with either a DC or an AC voltage. 

---