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These secularfrequencies are given by Equations (27) and (28):wu,n D n C 12 bu ,0n<1.27/1<n<0.28/n C 12 bu ,wu,n D11QUADRUPOLE ION TRAP MASS SPECTROMETERC .q2u /au.bu.q2u /2/2au.q2u /..bu4/2.bu26/auÐÐÐ.30/The resemblance of the simulated ion trajectory shownin Figure 6 to a roller coaster ride is due to the motionof an ion on the potential surface shown in Figure 3.
Theoscillatory motion of the ion results from the undulationsof the potential surface that can be envisaged as rotationof the potential surface. The simulation of the iontrajectory was carried out using the ITSIM (ion trajectorysimulation program),.17/ while the potential surface wasgenerated.18/ from Equation (24) by calculating fr,z forC20 D 1 and all of the other coefficients equal to zerofor increment steps of 1 mm in both radial and axialdirections.8 RESONANT EXCITATIONFigure 10 Stability diagram in (az , qz ) space for the region ofsimultaneous stability in both the r- and z-directions near theorigin for the three-dimensional quadrupole ion trap; the iso-brand iso-bz lines are shown in the diagram.
The qz -axis intersectsthe bz D 1 boundary at qz D 0.908, which corresponds to qmax inthe mass-selective instability mode.wheresbu ³q2au C u2.29/for qr < 0.2 and qz < 0.4. It should be noted that whilethe fundamental axial secular frequency, wz,0 , is usuallygiven in units of hertz in the literature and referred tosimply as wz , it should be given in radians per second.At this time, the higher-order frequencies are of littlepractical significance.It should be noted further that the definition ofbz given in Equation (29) above is only an approximation, known as the Dehmelt approximation afterHans Dehmelt who shared the 1989 Nobel Prize inPhysics along with Norman Ramsey and WolfgangPaul; bu is defined precisely by a continued fraction expression in terms of au and qu , as shown inEquation (30).22.bu C 2/2 au .q2u /.b2u C 4/2bu D au C .qu /au.q2u /2.bu C 6/auÐÐÐAs the motion of ions confined in a quadrupole ion trap ischaracterized by two secular frequencies, axial and radial,ion motion can be excited upon resonant irradiation ateither or both of these frequencies.
Such irradiation canbe effected by applying a small supplementary oscillating potential of some hundreds of millivolts across theend-cap electrodes, that is, in dipolar mode. Resonantexcitation, using the axial secular frequencies of confinedions, has become a powerful technique in quadrupole iontrap MS due to the utilization of predetermined waveforms composed of specified frequencies or frequencyranges. Prior to resonant excitation of ion trajectories,ions are focused collisionally to the vicinity of the center of the ion trap under the influence of collisions withhelium buffer gas atoms. This process is described as‘‘ion cooling’’ in that ion kinetic energies are reduced toca.
0.1 eV, corresponding to ca. 800 K as calculated from3RT/2 D 0.1 eV. Ion excursions from the center of theion trap are less than 1 mm.Resonant excitation of cooled ions, brought about bya supplementary potential oscillating at the axial secularfrequency of a specific ion species and of amplitude buta few hundreds of millivolts, causes those ions to moveaway from the ion trap center in such a way that theyexperience a greater trapping field. This process of ionexcitation is often referred to as ‘‘tickling’’. The ions areaccelerated further by the trapping field so that they canachieve kinetic energies of tens of electron volts.Resonant excitation is used in the following ways:1.to remove unwanted ions during ionization so asto isolate a narrow range of mass/charge ratios; inthis case, wavebands of frequencies are applied to12MASS SPECTROMETRYthe end-cap electrodes to excite and eject many ionspecies simultaneously leaving a single ion species (ora small range of mass/charge ratios) isolated withinthe ion trap;(corresponding to an electrode spacing, 2z0 , of 15.66 mm)and under the following conditions:U D 0;V D 757 V.0p/at 1.05 MHz2.to increase ion kinetic energy so as to promoteendothermic ion/molecule reactions; D 2pf D 2p ð 1.05 ð 106 rad s3.to increase ion kinetic energy so as to depositinternal energy in ions through momentum-exchangecollisions with helium atoms; in the limit, ionsdissociate.
This mode of resonant excitation isdiscussed in more detail in section 10.2 on CID;mD4.to increase kinetic energy so as to move ions closeto an end-cap electrode where an image current canbe detected. This mode permits the nondestructivemeasurement and remeasurement of the mass/chargeratio of confined ions;5.to increase kinetic energy so as to cause ions to escapefrom the trapping potential and be ejected.
This modecan be used either to eject unwanted ions, as in ionisolation, or to eject ions mass selectively while theapplied frequency is swept;6.to eject ions while the amplitude V of the main rfpotential is being ramped up. This mode, knownas axial modulation, is used in conjunction with anrf ramp so that ions come into resonance with afixed frequency of ca. 6 V.p p/ (peak-to-peak) justbefore their trajectories are made unstable. In thiscase, ions of low mass/charge ratio are removedfrom the perturbing influences of ions of highermass/charge ratio and are detected with enhancedresolution.In axial modulation, the resonant frequency is justless than half the main drive frequency . Resonantexcitation at lower frequencies has been used with greatsuccess to extend the normal mass range of the iontrap.9 CALCULATIONS134 ð 10 3 kg mol 1134 DaDAvogadro’s number6.022 ð 1023 mol 19.1 qz and Low-mass Cut-offFrom Equation (26), we recall thatqz Dm.r028eVC 2z20 /2.26/Thusqz D 8.1.602 ð 10C/.757 kg m2 s 2 C 1 /..134 ð 10 3 kg mol 1 /ð .6.022 ð 1023 mol 1 /ð .[1.000 C 1.226]10 4 m2 /.2p ð 1.05 ð 106 s 1 /219D 0.450We have now calculated that m/z 134 has a qz value of0.450 under these conditions, but what is the LMCO valueat qz slightly less than 0.908? Since m ð qz D constant atconstant V from Equation (26), the LMCO value can becalculated as:(LMCO).0.908/ D .m/z 134/.0.450/Rearranging,LMCO D .m/z 134/.0.450//.0.908/D m/z 66.4That is, with a potential of 757 V.0 p/ applied to the ringelectrode, only those ions of m/z > 66.4 will be stored.The potential V to be applied to the ring electrode toeffect a given LMCO is given asV D .LMCO/ ð .757 V.0On many occasions while working with a quadrupole iontrap it becomes necessary to calculate some of the iontrapping parameters such as qz , LMCO value (see below),bz , the secular frequency wz and the potential well depthDz .
In modern ion trap instruments, these calculationscan be carried out using the accompanying software butit is instructive to examine the manner in which each ofthe following parameters is calculated.Let us consider an ion of butylbenzene (m/z 134)in a normal stretched ion trap which has a ringelectrode of radius r0 D 1.00 cm and with z0 D 0.783 cm1D .11.40 ð LMCO/V.0p/ //.m/z66.4/p/This calculation is particularly useful when an ion is to befragmented and one wishes to know the low mass/chargelimit for fragment ions stored, that is, the LMCO.9.2 bzFromp Equation (29), we see that bz is given approximatelyby .q2z /2/ thus, when qz D 0.450, bz D 0.318. However,we have exceeded the limit of the approximation relating13QUADRUPOLE ION TRAP MASS SPECTROMETERqz and bz and so the calculated value of bz is high byabout 5%.
For m/z 1340, where qz D 0.0450 such that theabove approximation is valid, the value of bz D 0.0318.9.3 !zFrom Equation (27), the fundamental axial secular frequency, wz , (or, more properly, wz,0 ) is given by bz /2thus, when bz D 0.318 and D 2p ð 1.05 ð 106 rad s 1 ,wz D 1.049 ð 106 rad s 1 or, more conventionally, wz D167 kHz; wz is correspondingly high by about 5%. However, for m/z 1340, wz is 16.7 kHz.9.4 Mass RangeThe upper limit of the mass range is given by themass/charge ratio having a qz value of, let us say, exactly0.900 when the maximum rf amplitude is applied tothe ring electrode.
From Equation (26) it is seen thatm ð qz /V D constant; this constant can be evaluated fromthe above expression for qz as 0.0797. With qz D 0.900 andV D 7340 V.0 p/ , the mass range is found to be 650 Da.9.5 Mass Range ExtensionWith V D 7340 V.0 p/ and ion ejection brought about byaxial modulation at, say, qz D 0.900, the upper mass limitof the ion trap is 650 Da; that is, for m/z 650, qz D 0.900.However, under these trapping conditions, ions of m/z1300 remain stored in the ion trap and have a qz valueof 0.450.
If resonant excitation had been carried out atqz D 0.45 with excitation at 167 kHz (see above), ionsof m/z 1300 would be on the point of ejection at themaximum rf amplitude and the mass range of the iontrap would have been doubled to 1300 Da. Similarly,with resonant excitation at qz D 0.045, the mass rangewould be extended to 13 000 Da. This method has beenapplied.19/ with great success, using axial modulation atlow qz and a slightly lower drive frequency , such that themass/charge range was extended to 72 000 Da per charge.for a given mass, mass resolution was increased.
While inresearch instruments peak widths of some 20 µDa havebeen observed, the narrowest peak widths in commercialinstruments are ca. 0.2 Da so that, for m/z 2000, a massresolution approaching 10 000 is achieved.9.7 DzThe essential importance of the potential well depth isthat it determines both the minimum kinetic energy thatan ion must acquire by resonant excitation in order tobe ejected from the ion trap and the maximum kineticenergy that an externally generated ion may possess andstill be trapped.
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