quadr-iontrap1 (1248334), страница 4
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The expression for qz in Equation (17) contains themass/charge ratio for a given ion, the size of the iontrap, r0 , the amplitude V of the rf potential and theradial frequency ; these parameters are all that we shallneed in order to understand the various operations ofthe ion trap. The solutions to the Mathieu equation canbe calculated in terms of az and qz and these solutionscan be interpreted in terms of trajectory stability (andinstability) in each of the x, y (or r), and z directions.
Whenthe confinement conditions correspond simultaneously totrajectory stability in both r- and z-directions, a chargedparticle may be stored. The trajectories of a chargedparticle are characterized by the fundamental secularfrequencies of ion motion in the radial, r, and axial, z,directions.6.3 Potentials on the ElectrodesThe potentials on the ring and end-cap electrodes maybe verified in the following manner with reference toFigure 2 and Equation (12).
Consider the intersectionof the central radial plane with the surface of the ringelectrode, such that z D 0 and r D r0 ; the potential at thering electrode is given by Equation (18).15/We can now compare directly the terms on the right handsides of Equations (6) and (15), recalling that u representsx, to obtain, Equation (16)ax Dthat qx D qy ; this equality is found since l D s D 1.
Itis suggested that it would be instructive for the readerto derive similarly the az and qz trapping parametersfor positively charged ions (Equation 17) which are usedfrequently in discussions of the stability diagram of thequadrupole ion trap; in this case, l D s D 1 and g D 2.For negatively charged ions, the signs of the az and qztrapping parameters in Equation (17) are reversed:.16/When this derivation is repeated to obtain the force onan ion in the y-direction in a quadrupole mass filter, onefinds that qx D qy ; this relationship is obtained sincel D s D 1. For the quadrupole ion trap, it is foundfr0 ,0 Df0 2r D f0r02 0.18/Now consider the intersection of the central axis ofcylindrical symmetry with the surface of either end-capelectrode, such that r D 0 and z D z0 ; recalling the identityof Equation (3) and the potential of Equation (12),the potential at each end-cap electrode is given byEquation (19)f0,z0 Df0.
2z20 / Dr02f0.19/9QUADRUPOLE ION TRAP MASS SPECTROMETERHowever, no commercial quadrupole ion trap is operatedin this fashion; rather, the end-cap electrodes are held atground potential (except for the imposition of oscillatingpotentials of low amplitude, hundreds of millivolts toa few volts). The net effect of applying f0 to the ringelectrode and grounding the end-cap electrodes is to halvethe mass range of the ion trap as a mass spectrometer.In order to verify the potentials on the ring and endcap electrodes in the commercial ion trap, an alternativeequation to Equation (12) must be used; this equation isf0 .r2 2z2 /Cc2r02fr.z D.20/where c is a constant. The ring electrode potential (Equation (20) with z D 0, r D r0 ) is given by Equation (21)fr0 ,0f0 r02DC c D f02r02f0,z0 Df0 .r2 2z2 / f0C22r02m.r028eU;C 2z20 /2qr Dm.r0216eU;C 2z20 /2qz Dm.r024eVC 2z20 /2.25/m.r028eVC 2z20 /2.26/and.22/Hence Equation (20) reduces to Equation (23)fr,z Dar D.21/from which we obtain c D f0 /2.
The potential at theend-cap electrodes (r D 0 and z D z0 ) is given byEquation (22)2f0 z20f0D0C222r0In order to compensate for these higher order multipole components, the electrodes of most commercialinstruments were assembled in such a way that the distance between the end-cap electrodes was increased or‘‘stretched’’; the value of z0 was increased by 10.6%.However, there was no corresponding modification of theshapes of the electrodes which would be required in orderto maintain a purely quadrupolar geometry.The immediate consequences are that the asymptotesto the end-cap electrodes no longer coincide with thosefor the ring electrode.
Furthermore, r02 6D 2z20 . In order tocompensate, in part, for the stretching of the ion trap, thetrapping parameters are now calculated using the actualvalues of z0 and r0 , as follows, Equation (25).23/The constant term does not change the equations ofmotion derived from the partial differentials, but thepotential along the asymptotes of the hyperbolas ischanged. It should be noted that an ion at the centerof a commercial ion trap experiences a potential of f0 /2and ‘‘sees’’ a potential of f0 /2 on the end-cap electrodesand a potential of f0 /2 on the ring electrode.6.4 Stretched Ion TrapAs we discussed above, the electrodes of the ion trapare truncated in order to obtain a practical workinginstrument but this truncation introduces higher ordermultipole components to the potential as shown inEquation (24)fr,z D C00 C C10 z C C20 12 r2 z2 C C30 z 32 r2 z2.24/C C40 38 r4 3r2 z2 C z4 C Ð Ð ÐThe Cn0 coefficients, where n D 0, 1, 2, 3, and 4, correspondto monopole, dipole, quadrupole, hexapole and octopolecomponents, respectively, of the potential fr,z .
Forthe pure quadrupole ion trap, only the coefficientscorresponding to n D 0 and n D 2 are nonzero.az DWhen r02 D 2z20 (Equation 3) is substituted into Equation (26), we obtain the trapping parameters given inEquation (17). It should be noted that for the ion trapin the LCQ and GCQ instruments, r0 D 0.707 cm andz0 D 0.785 cm such that the geometry has been stretchedby ca. 57%.7 REGIONS OF ION TRAJECTORYSTABILITYThe operation of the quadrupole ion trap is concernedwith the criteria that govern the stability (or instability)of the trajectory of an ion in the ion trap, that is, theexperimental conditions that determine whether an ionis stored within the device or whether it is ejected andwhether it is excited resonantly or not.
The experimentalconditions for the manipulation of ion trajectories arerepresented mathematically by the solutions to Mathieu’sequation.The solutions to Mathieu’s equation are of two types:(i) ion motion is periodic but unstable, and (ii) ionmotion is periodic and stable.
Solutions of type (i) arecalled Mathieu functions of integral order and form theboundaries of unstable regions on the stability diagram.The boundaries, which are referred to as characteristiccurves or characteristic values, correspond to the valuesof a new trapping parameter, bz , that is, 0, 1, 2, 3, .
. .; bz isa complex function of az and qz to which we shall return.The boundaries represent, in practical terms, the point atwhich the trajectory of an ion becomes unbounded.10MASS SPECTROMETRY10z-stable5Bauz-stable051015A−5r-stable−10quFigure 9 Mathieu stability diagram in (az , qz ) space for thequadrupole ion trap in both the r- and z-directions. Regions ofsimultaneous overlap are labeled A and B.Figure 8 Several Mathieu stability regions for the three-dimensional quadrupole field. (a) diagrams for the z-direction of(az , qz ) space; (b) diagrams for the r-direction of (az , qz ) space.Solutions of type (ii) determine the motion of ions inan ion trap. The stability regions corresponding to stablesolutions of the Mathieu equation in the z-directionare shaded and labeled ‘‘z-stable’’ in Figure 8(a). Thestability regions corresponding to stable solutions ofthe Mathieu equation in the r-direction are shaded andlabeled ‘‘r-stable’’ in Figure 8(b); it can be seen that theyare doubled in magnitude along the ordinate and inverted.It is seen from Equations (25) and (26) that az D 2ar andqz D 2qr , that is, the stability parameters for the r- andz-directions differ by a factor of 2.7.1 Stability RegionIons can be stored in the ion trap provided thattheir trajectories are stable in the r- and z-directionssimultaneously; such trajectory stability is obtained in theregion closest to the origin, that is, region A in Figure 9which is plotted in au , qu space, that is, where au is plottedagainst qu .
Regions A and B are referred to as stabilityregions; region A is of the greatest importance at this time(region B remains to be explored) and is shown in greaterdetail in Figure 10. The coordinates of the stability regionin Figure 10 are the Mathieu parameters az and qz . Here,we plot az versus qz rather than au versus qu which isunnecessarily confusing because u D r, z. In Figure 10,the bz D 1 stability boundary intersects with the qz axis atqz D 0.908; this working point is that of the ion of lowestmass/charge ratio (that is, low-mass cut-off (LMCO), asdiscussed below) which can be stored in the ion trap.The stability diagram for negatively charged ions is themirror image about the qz -axis of the stability diagram inFigure 10.7.2 Secular FrequenciesA three-dimensional representation of an ion trajectory inthe ion trap, as shown in Figure 6, has the general appearance of a Lissajous curve or figure-of-eight composed oftwo fundamental frequency components, radial and axial,respectively, wr,0 and wz,0 of the secular motion..1/ Thedescription of ‘‘fundamental’’ infers that there exist otherhigher order (n) frequencies and that the entire family offrequencies is described by wr,n and wz,n .
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