quadr-iontrap1 (1248334), страница 3
Текст из файла (страница 3)
To obviate this problem, ions of lowestmass/charge ratio are made to come into resonance witha fixed frequency applied across the end-cap electrodes.Ions come into resonance as the rf amplitude is ramped,that is, as the amplitude is increased. As the ions becomeexcited resonantly in the axial direction, they emergefrom the onion and experience a brief period free of spacecharge immediately prior to ejection. During ejection, theions remain clustered together and, upon impacting on thedetector, produce ion signals of higher mass resolution.This process of ion axial excitation prior to ion ejection istermed axial modulation.Let us consider an analogy to this trapping, focusing,and ejection process.
The trapping pseudo-potential wellcreated within the electrode assembly of the quadrupoleion trap can be likened to a bowl or glass of paraboliccross-section; ion species are confined in layers in the bowlrather like an exotic drink of several liqueurs arrangedcarefully and tastefully in horizontal layers according totheir density, as shown in Figure 7.
The liqueur of greatestdensity represents the ions of lowest mass/charge ratio.The tilting of the bowl or the lowering of the side of thebowl corresponds to the ramping of the rf amplitude; theliqueur glass in Figure 7 corresponds to the detector. Theremoval of ions from near the bottom of the potentialwell is effected with the straw in Figure 7. The tiltingof the bowl is analogous to ramping the rf amplitudewhile the use of a straw is analogous to axial modulationin which ions are brought successively into resonancewith an applied AC (alternating current) frequency, as isdiscussed later. As the liqueurs are drawn up the strawin order of decreasing density, so the ions are ejected inorder of increasing mass/charge ratio and impinge upona detector. The signals from the detector create a massspectrum of the contents of the ion trap.5 STRUCTURE OF THE QUADRUPOLE IONTRAPThe quadrupole ion trap is composed of three electrodeswith the ring electrode located symmetrically betweentwo end-cap electrodes as shown in Figure 2.
The shapeof the ring electrode is given by Equation (1)r2r02z2D1z20.1/while the shapes of the end-cap electrodes are given byEquation (2)r2r02z2Dz201.2/For an ideal quadrupole field, the following identity isgiven (usually with more authority than truth)r02 D 2z20.3/so that, once the magnitude of r0 is given, the sizes of allthree electrodes and the spacing between the electrodesare fixed.
However, it has been pointed out by Knight.15/that, contrary to Equation (3), the ratio of z20 : r02 is notnecessarily restricted to 2. Regardless of the value of thisratio, the size of the ion trap, is determined largely by themagnitude of r0 and, in the majority of commercial iontraps in use today, r0 is either 1.00 cm or 0.707 cm.(a)(b)Figure 7 Schematic presentation of (a) a trapping parabolicpotential well where the three liquids differing in densityrepresent ions differing in mass/charge ratio; (b) the tiltingof the well corresponds to ramping of the rf potential whilethe straw, with which ions are withdrawn in order of increasingmass/charge ratio, represents axial modulation.6 THEORY OF QUADRUPOLE ION TRAPOPERATIONThe motions of ions in quadrupole devices differsgreatly from the straight lines and arcing curves of ionsin field-free regions and in magnetic and electrostaticsectors, respectively, familiar to those conversant withsector mass spectrometers.
The quadrupole ion trap and7QUADRUPOLE ION TRAP MASS SPECTROMETERthe quadrupole mass filter or analyzer are describedas dynamic instruments since ion trajectories in theseinstruments are influenced by a set of time-dependentforces (which render the trajectories mathematically moredifficult to predict compared with sector instruments).Sector instruments are described as static devices in thatthe field is maintained at a constant value for transmissionof an ion. In quadrupole instruments, a quadrupole fieldis established when a potential is applied to electrodesthat have a hyperbolic geometric form. Let us examinethe movement of charged particles in a quadrupole fieldby considering first the forces acting on a single ion withina quadrupole field.where u represents the coordinate axes x, y and z, x isa dimensionless parameter equal to t/2 such that must be a frequency as t is time, and au and qu areadditional dimensionless parameters known as trappingor stability parameters.
The introduction of here isnot entirely serendipitous since it will reappear as theradial frequency (in rad s 1 ) of the rf potential applied tothe ring electrode. Now it can be shown by substitutingx D t/2 (from Equation 4) thatd2 u2 d2 uDdt24 dx2Substitution of Equation (5) into Equation (4), multiplying throughout by m and rearranging yields6.1 An Ion in a Quadrupole FieldmAn ion, positively charged or negatively charged, in aquadrupole field experiences strong focusing in that therestoring force, which drives the ion back towards thecenter of the device, increases as the ion deviates from thecenter of the device. The motion of ions in a quadrupolefield can be described mathematically by the solutionsto the second-order linear differential equation describedoriginally by Mathieu;.16/ this equation is known as theMathieu equation.
From Mathieu’s investigation of themathematics of vibrating stretched skins, he was able todescribe solutions in terms of regions of stability andinstability; these solutions and the criteria for stabilityand instability have been used to describe the trajectoriesof ions confined in quadrupole devices and to definethe limits to trajectory stability. In order to adopt thesolutions to the Mathieu equation, we must verify thatthe equation of motion of an ion confined in a quadrupoledevice can be described by the Mathieu equation. Thepath that is followed here concerns the expression for aforce (mass ð acceleration) in Mathieu’s equation, andcomparison of that expression with one for the force onan ion in a quadrupole field.
This comparison is laid outbelow in simple mathematical terms; thus it is possible toexpress the magnitudes and frequencies of the potentialsapplied to ion trap electrodes, the size of the ion trap, andthe mass/charge ratio of ions confined therein in terms ofMathieu’s dimensionless parameters, au and qu . On thisbasis, we shall adopt the idea of stability regions in au , quspace in order to discuss the trapping, and limits thereto,of gaseous ions in quadrupole devices.6.2 The Mathieu EquationThe canonical or commonly accepted form of the Mathieuequation is, Equation (4):d2 uC .audx22qu cos 2x/u D 0.4/.5/d2 uDdt2m2.au42qu cos t/u.6/We note that the left side of Equation (6) can representthe force on an ion, that is, mass times acceleration ineach of the x, y and z directions.Now the field in quadrupole devices is uncoupled sothat the forces in the three coordinate directions maybe determined separately.
Let us then consider the forcein the x-direction, Fx , experienced by an ion of massm and charge e at any point within a quadrupole field,Equation (7)Fx D ma D md2 xDdt2e@f@x.7/where a is the acceleration of the ion, e is the electroniccharge and f is the potential at any point (x,y,z) within thefield. Similar expressions for Fy and Fz can be obtained.It should be noted that Equation (7) relates the force onan ion to the field within the ion trap.
The quadrupolepotential f can be expressed asfDf0.lx2 C sy2 C gz2 /r02.8/where f0 is the applied electric potential (which weshall see later is an rf potential either alone or incombination with a direct current (DC) potential), l,s and g are weighting constants for the x, y, and zcoordinates, respectively, and r0 is a constant which isdefined separately depending on whether the quadrupoledevice is an ion trap or mass filter. It can be seen fromEquation (8) that the potential increases quadraticallywith x, y, and z.
In any electric field, it is essential thatthe Laplace condition, which requires that the seconddifferential of the potential at a point be equal to zero,be satisfied; the Laplace condition ensures that the fieldin the x, y, and z directions is linear and does not change.When that is done, it is found that, Equation (9)lCsCgD0.9/8MASS SPECTROMETRYFor the ion trap, l D s D 1 and g D 2, whereas for thequadrupole mass filter l D s D 1 and g D 0.
Substitutingthe values l D s D 1 and g D 2 into Equation (8), weobtain Equation (10) for the potential at any point withinthe quadrupole field in a quadrupole ion trap.fx,y,z Df0 2.x C y2r022z2 /.10/This equation can be transformed into cylindrical coordinates by employing the standard transformationsx D r cos q, y D r sin q, z D z.
Thus Equation (10) becomesEquation (11)fr,z Df0 2.r cos2 q C r2 sin2 qr022z2 /.11/When we apply the trigonometric identity cos2 C sin2 D 1,we obtain Equation (12)fr,z Df0 2.rr022z2 /.12/The applied electric potential, f0 , (that is, applied tothe ring electrode) is either an rf potential V cos tor a combination of a DC potential, U, of the form,Equation (13)f0 D U C V cos t.13/where is the angular frequency (in rad s 1 ) of the rffield. Note that is equal to 2pf , where f is the frequencyin hertz.When the expression for f0 as given by Equation (13)and l D 1 are substituted into Equation (8) and f isdifferentiated with respect to x, Equation (14) is obtainedfor the potential gradient2x@fD 2 .U C V cos t/@xr0.14/Substitution of Equation (14) into Equation (7) yields anexpression for the force on an ion, Equation (15)md2 x2eD 2 .U C V cos t/x2dtr08eU;mr02 2qx D4eVmr02 2az D8eU;mr02 2qz D4eVmr02 2.17/Let us ignore az (which is proportional to U, a DCpotential) for the present since most commercial ion trapinstruments do not offer the flexibility of applying a DCpotential to the electrodes; therefore, az is held equalto zero such that the most common mode of ion trapoperation is said to correspond to operation on the qzaxis.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
